Dpp- to MATHEMATICS Dail Practice Problems Target IIT JEE 00 CLASS : XIII (VXYZ) DPP. NO.- to DPP- Q. If on a given base, a triangle be described such that the sum of the tangents of the base angles is a constant, then the locus of the vertex is : a circle (B) a parabola (C) an ellipse (D a hperbola Q. The locus of the point of trisection of all the double ordinates of the parabola = lx is a parabola whose latus rectum is 9 l (B) l 9 (C) 9 l l (D) 6 Q. Let a variable circle is drawn so that it alwas touches a fixed line and also a given circle, the line not passing through the centre of the circle. The locus of the centre of the variable circle, is a parabola (B) a circle (C) an ellipse (D) a hperbola Q. The vertex A of the parabola = ax is joined to an point P on it and PQ is drawn at right angles to AP to meet the axis in Q. Projection of PQ on the axis is equal to twice the latus rectum (B) the latus rectum (C) half the latus rectum (D) one fourth of the latus rectum Q.5 Two unequal parabolas have the same common axis which is the x-axis and have the same vertex which is the origin with their concavities in opposite direction. If a variable line parallel to the common axis meet the parabolas in P and P' the locus of the middle point of PP' is a parabola (B) a circle (C) an ellipse (D) a hperbola Q.6 The straight line = m(x a) will meet the parabola = ax in two distinct real points if m R (B) m [, ] (C) m (, ] [, )R (D) m R {0} x Q.7 All points on the curve = a x + a sin at which the tangent is parallel to x-axis lie on a a circle (B) a parabola (C) an ellipse (D) a line Q.8 Locus of trisection point of an arbitrar double ordinate of the parabola x = b, is 9x = b (B) x = b (C) 9x = b (D) 9x = b Q.9 The equation of the circle drawn with the focus of the parabola (x ) 8 = 0 as its centre and touching the parabola at its vertex is : x + = 0 (B) x + + = 0 (C) x + x = 0 (D) x + x + = 0 Q.0 The length of the latus rectum of the parabola, 6 + 5x = 0 is (B) (C) 5 (D) 7 Q. Which one of the following equations represented parametricall, represents equation to a parabolic profile? x = cos t ; = sin t (B) x = cos t ; = cos t (C) x = tan t ; = sec t (D) x = sin t ; = sin t + cos t Dpp's on Conic Section (Parabola, Ellipse, Hperbola) []
Q. The length of the intercept on axis cut off b the parabola, 5 = x 6 is (B) (C) (D) 5 Q. A variable circle is described to pass through (, 0) and touch the line = x. The locus of the centre of the circle is a parabola, whose length of latus rectum, is (B) (C) (D) Q. Angle between the parabolas = b (x a + b) and x + a ( b a) = 0 at the common end of their latus rectum, is tan () (B) cot + cot + cot (C) tan ( ) (D) tan () + tan () Q.5 A point P on a parabola = x, the foot of the perpendicular from it upon the directrix, and the focus are the vertices of an equilateral triangle, find the area of the equilateral triangle. Q.6 Given = ax + bx + c represents a parabola. Find its vertex, focus, latus rectum and the directrix. Q.7 Prove that the locus of the middle points of all chords of the parabola = ax passing through the vetex is the parabola = ax. Q.8 Prove that the equation to the parabola, whose vertex and focus are on the axis of x at distances a and a' from the origin respectivel, is = (a' a)(x a). Q.9 Prove that the locus of the centre of a circle, which intercepts a chord of given length a on the axis of x and passes through a given point on the axis of distant b from the origin, is the curve Q.0 A variable parabola is drawn to pass through A & B, the ends of a diameter of a given circle with centre at the origin and radius c & to have as directrix a tangent to a concentric circle of radius 'a' (a >c) ; the axes being AB & a perpendicular diameter, prove that the locus of the focus of the parabola is the standard ellipse + = where b = a c. DPP- Q. If a focal chord of = ax makes an angle α, α (0, /] with the positive direction of x-axis, then minimum length of this focal chord is 6a (B) a (C) 8a (D) None Q. OA and OB are two mutuall perpendicular chords of = ax, 'O' being the origin. Line AB will alwas pass through the point (a, 0) (B) (6a, 0) (C) (8a, 0) (D) (a, 0) Q. ABCD and EFGC are squares and the curve = k x passes through FG the origin D and the points B and F. The ratio is BC 5 + + 5 + + (B) (C) (D) Q. From an external point P, pair of tangent lines are drawn to the parabola, = x. If θ & θ are the inclinations of these tangents with the axis of x such that, θ + θ =, then the locus of P is : x + = 0 (B) x + = 0 (C) x = 0 (D) x + + = 0 Dpp's on Conic Section (Parabola, Ellipse, Hperbola) []
Q.5 Maximum number of common chords of a parabola and a circle can be equal to (B) (C) 6 (D) 8 Q.6 PN is an ordinate of the parabola = ax. A straight line is drawn parallel to the axis to bisect NP and meets the curve in Q. NQ meets the tangent at the vertex in apoint T such that AT = knp, then the value of k is (where A is the vertex) / (B) / (C) (D) none Q.7 Let A and B be two points on a parabola = x with vertex V such that VA is perpendicular to VB and VA θ is the angle between the chord VA and the axis of the parabola. The value of is VB tan θ (B) tan θ (C) cot θ (D) cot θ Q.8 Minimum distance between the curves = x and x = is equal to 5 7 (B) (C) (D) Q.9 The length of a focal chord of the parabola = ax at a distance b from the vertex is c, then a = bc (B) a = b c (C) ac = b (D) b c = a Q.0 The straight line joining an point P on the parabola = ax to the vertex and perpendicular from the focus to the tangent at P, intersect at R, then the equaiton of the locus of R is x + ax = 0 (B) x + ax = 0 (C) x + a = 0 (D) x + a = 0 Q. Locus of the feet of the perpendiculars drawn from vertex of the parabola = ax upon all such chords of the parabola which subtend a right angle at the vertex is x + ax = 0 (B) x + ax = 0 (C) x + + ax = 0 (D) x + + ax = 0 More than one are correct: Q. Consider a circle with its centre ling on the focus of the parabola, = px such that it touches the directrix of the parabola. Then a point of intersection of the circle & the parabola is p p, p (B), p (C) p p, (D) p p, DPP- Q. -intercept of the common tangent to the parabola = x and x = 08 is 8 (B) (C) 9 (D) 6 Q. The points of contact Q and R of tangent from the point P (, ) on the parabola = x are (9, 6) and (, ) (B) (, ) and (, ) (C) (, ) and (9, 6) (D) (9, 6) and (, ) Q. Length of the normal chord of the parabola, = x, which makes an angle of with the axis of x is: 8 (B) 8 (C) (D) Q. If the lines ( b) = m (x + a) and ( b) = m (x + a) are the tangents to the parabola = ax, then m + m = 0 (B) m m = (C) m m = (D) m + m = Q.5 If the normal to a parabola = ax at P meets the curve again in Q and if PQ and the normal at Q makes angles α and β respectivel with the x-axis then tan α (tan α + tan β) has the value equal to 0 (B) (C) (D) Dpp's on Conic Section (Parabola, Ellipse, Hperbola) []
Q.6 C is the centre of the circle with centre (0, ) and radius unit. P is the parabola = ax. The set of values of 'a' for which the meet at a point other than the origin, is a > 0 (B) a 0, (C), (D), Q.7 PQ is a normal chord of the parabola = ax at P, A being the vertex of the parabola. Through P a line is drawn parallel to AQ meeting the x axis in R. Then the length of AR is : equal to the length of the latus rectum (B) equal to the focal distance of the point P (C) equal to twice the focal distance of the point P (D) equal to the distance of the point P from the directrix. Q.8 Length of the focal chord of the parabola = ax at a distance p from the vertex is : a p (B) a (C) a p p (D) p a Q.9 The triangle PQR of area 'A' is inscribed in the parabola = ax such that the vertex P lies at the vertex of the parabola and the base QR is a focal chord. The modulus of the difference of the ordinates of the points Q and R is : A a (B) A a (C) A a (D) A a Q.0 The roots of the equation m m + 5 = 0 are the slopes of the two tangents to the parabola = x. The tangents intersect at the point, (B), (C), 5 5 5 5 5 5 (D) point of intersection can not be found as the tangents are not real Q. Through the focus of the parabola = px (p > 0) a line is drawn which intersects the curve at A(x, ) and B(x, ). The ratio equals xx (B) (C) (D) some function of p Q. If the line x + + K = 0 is a normal to the parabola, + 8x = 0 then K = 6 (B) 8 (C) (D) Q. The normal chord of a parabola = ax at the point whose ordinate is equal to the abscissa, then angle subtended b normal chord at the focus is : / (B) tan (C) tan (D) / Q. The point(s) on the parabola = x which are closest to the circle, x + + 8 = 0 is/are (0, 0) (B) (, ) (C) (, ) (D) none More than one are correct: Q.5 Let = ax be a parabola and x + + bx = 0 be a circle. If parabola and circle touch each other externall then : a > 0, b > 0 (B) a > 0, b < 0 (C) a < 0, b > 0 (D) a < 0, b < 0 Q.6 The straight line + x = touches the parabola : x + = 0 (B) x x + = 0 (C) x x + = 0 (D) x x + = 0 Dpp's on Conic Section (Parabola, Ellipse, Hperbola) []
DPP- Q. TP & TQ are tangents to the parabola, = ax at P & Q. If the chord PQ passes through the fixed point ( a, b) then the locus of T is : a = b (x b) (B) bx = a ( a) (C) b = a (x a) (D) ax = b ( b) Q. Through the vertex O of the parabola, = ax two chords OP & OQ are drawn and the circles on OP & OQ as diameters intersect in R. If θ, θ & φ are the angles made with the axis b the tangents at P & Q on the parabola & b OR then the value of, cot θ + cot θ = tan φ (B) tan ( φ) (C) 0 (D) cot φ Q. If a normal to a parabola = ax makes an angle φ with its axis, then it will cut the curve again at an angle tan ( tanφ) (B) tan tanφ (C) cot tanφ (D) none Q. Tangents are drawn from the points on the line x + = 0 to parabola = 8x. Then the variable chords of contact pass through a fixed point whose coordinates are : (, ) (B) (, ) (C) (, ) (D) (, ) Q.5 If the tangents & normals at the extremities of a focal chord of a parabola intersect at (x, ) and (x, ) respectivel, then : x = x (B) x = (C) = (D) x = Q.6 If two normals to a parabola = ax intersect at right angles then the chord joining their feet passes through a fixed point whose co-ordinates are : ( a, 0) (B) (a, 0) (C) (a, 0) (D) none Q.7 The equation of a straight line passing through the point (, 6) and cutting the curve = x orthogonall is x + 8 =0 (B) x + 9 = 0 (C) x 6 = 0 (D) none Q.8 The tangent and normal at P(t), for all real positive t, to the parabola = ax meet the axis of the parabola in T and G respectivel, then the angle at which the tangent at P to the parabola is inclined to the tangent at P to the circle passing through the points P, T and G is cot t (B) cot t (C) tan t (D) tan t Q.9 A circle with radius unit has its centre on the positive -axis. If this circle touches the parabola = x tangentiall at the points P and Q then the sum of the ordinates of P and Q, is 5/ (B) 5/8 (C) 5 (D) 5 Q.0 Normal to the parabola = 8x at the point P (, ) meets the parabola again at the point Q. If C is the centre of the circle described on PQ as diameter then the coordinates of the image of the point C in the line = x are (, 0) (B) (, 8) (C) (, 0) (D) (, 0) Q. Two parabolas = a(x - l ) and x = a ( l ) alwas touch one another, the quantities l and l are both variable. Locus of their point of contact has the equation x = a (B) x = a (C) x = a (D) none Q. A pair of tangents to a parabola is are equall inclined to a straight line whose inclination to the axis is α. The locus of their point of intersection is : a circle (B) a parabola (C) a straight line (D) a line pair Q. In a parabola = ax the angle θ that the latus rectum subtends at the vertex of the parabola is dependent on the length of the latus rectum (B) independent of the latus rectum and lies between 5 6 & (C) independent of the latus rectum and lies between & 5 6 (D) independent of the latus rectum and lies between & Dpp's on Conic Section (Parabola, Ellipse, Hperbola) [5]
DPP-5 Q. Normals are drawn at points A, B, and C on the parabola = x which intersect at P(h, k). The locus of the point P if the slope of the line joining the feet of two of them is, is x + = (B) x = (C) = (x ) (D) = x Q. Tangents are drawn from the point (, ) on the parabola = x. The length, these tangents will intercept on the line x = is : 6 (B) 6 (C) 6 (D) none of these Q. Which one of the following lines cannot be the normals to x =? x + = 0 (B) x + = 0 (C) x + = 0 (D) x + + = 0 x Q. An equation of the line that passes through (0, ) and is perpendicular to = is x + = 9 (B) x + = 9 (C) x + = 9 (D) x + = 8 Paragraph for question nos. 5 to 6 Consider the parabola = 8x Q.5 Area of the figure formed b the tangents and normals drawn at the extremities of its latus rectum is 8 (B) 6 (C) (D) 6 Q.6 Distance between the tangent to the parabola and a parallel normal inclined at 0 with the x-axis, is 6 6 6 (B) (C) (D) 9 Paragraph for question nos. 7 to 9 Tangents are drawn to the parabola = x from the point P(6, 5) to touch the parabola at Q and R. C is a circle which touches the parabola at Q and C is a circle which touches the parabola at R. Both the circles C and C pass through the focus of the parabola. Q.7 Area of the PQR equals / (B) (C) (D) / Q.8 Radius of the circle C is 5 5 (B) 5 0 (C) 0 (D) 0 Q.9 The common chord of the circles C and C passes through the incentre (B) circumcentre (C) centroid (D) orthocentre of the PQR Paragraph for question nos. 0 to Tangents are drawn to the parabola = x at the point P which is the upper end of latus rectum. Q.0 Image of the parabola = x in the tangent line at the point P is (x + ) = 6 (B) (x + ) = 8( ) (C) (x + ) = ( ) (D) (x ) = ( ) Q. Radius of the circle touching the parabola = x at the point P and passing through its focus is (B) (C) (D) Q. Area enclosed b the tangent line at P, x-axis and the parabola is (B) (C) (D) none Dpp's on Conic Section (Parabola, Ellipse, Hperbola) [6]
More than one are correct: Q. Let P, Q and R are three co-normal points on the parabola = ax. Then the correct statement(s) is/are algebraic sum of the slopes of the normals at P, Q and R vanishes (B) algebraic sum of the ordinates of the points P, Q and R vanishes (C) centroid of the triangle PQR lies on the axis of the parabola (D) circle circumscribing the triangle PQR passes through the vertex of the parabola Q. Variable chords of the parabola = ax subtend a right angle at the vertex. Then : locus of the feet of the perpendiculars from the vertex on these chords is a circle (B) locus of the middle points of the chords is a parabola (C) variable chords passes through a fixed point on the axis of the parabola (D) none of these Q.5 Through a point P (, 0), tangents PQ and PR are drawn to the parabola = 8x. Two circles each passing through the focus of the parabola and one touching at Q and other at R are drawn. Which of the following point(s) with respect to the triangle PQR lie(s) on the common chord of the two circles? centroid (B) orthocentre (C) incentre (D) circumcentre Q.6 TP and TQ are tangents to parabola = x and normals at P and Q intersect at a point R on the curve. The locus of the centre of the circle circumscribing TPQ is a parabola whose 7 vertex is (, 0). (B) foot of directrix is, 0. 8 9 (C) length of latus-rectum is. (D) focus is, 0. 8 Match the column: Q.7 Consider the parabola = x Column-I Column-II Tangent and normal at the extremities of the latus rectum intersect (P) (0, 0) the x axis at T and G respectivel. The coordinates of the middle point of T and G are (B) Variable chords of the parabola passing through a fixed point K on (Q) (, 0) the axis, such that sum of the squares of the reciprocals of the two parts of the chords through K, is a constant. The coordinate of the (C) point K are (R) (6, 0) All variable chords of the parabola subtending a right angle at the origin are concurrent at the point (D) AB and CD are the chords of the parabola which intersect at a point (S) (, 0) E on the axis. The radical axis of the two circles described on AB and CD as diameter alwas passes through Subjective: Q.8 Let L : x + = 0 and L : x = 0 are tangent to a parabola whose focus is S(, ). m If the length of latus-rectum of the parabola can be expressed as (where m and n are coprime) then find the value of (m + n). n Dpp's on Conic Section (Parabola, Ellipse, Hperbola) [7]
DPP-6 Q. Let 'E' be the ellipse x 9 + = & 'C' be the circle x + = 9. Let P & Q be the points (, ) and (, ) respectivel. Then : Q lies inside C but outside E (B) Q lies outside both C & E (C) P lies inside both C & E (D) P lies inside C but outside E. Q. The eccentricit of the ellipse (x ) + ( ) = 9 is (B) Q. An ellipse has OB as a semi minor axis where 'O' is the origin. F, F are its foci and the angle FBF is a right angle. Then the eccentricit of the ellipse i (B) (C) (C) Dpp's on Conic Section (Parabola, Ellipse, Hperbola) [8] (D) (D) Q. There are exactl two points on the ellipse + = whose distance from the centre of the ellipse are greatest and equal to (B) a + b. Eccentricit of this ellipse is equal to Q.5 A circle has the same centre as an ellipse & passes through the foci F & F of the ellipse, such that the two curves intersect in points. Let 'P' be an one of their point of intersection. If the major axis of the ellipse is 7 & the area of the triangle PF F is 0, then the distance between the foci is : (B) (C) (D) none Q.6 The latus rectum of a conic section is the width of the function through the focus. The positive difference between the lengths of the latus rectum of = x + x 9 and x + 6x + 6 = is / (B) (C) / (D) 5/ Q.7 Imagine that ou have two thumbtacks placed at two points, A and B. If the ends of a fixed length of string are fastened to the thumbtacks and the string is drawn taut with a pencil, the path traced b the pencil will be an ellipse. The best wa to maximise the area surrounded b the ellipse with a fixed length of string occurs when I the two points A and B have the maximum distance between them. II two points A and B coincide. III A and B are placed verticall. IV The area is alwas same regardless of the location of A and B. I (B) II (C) III (D) IV Q.8 An ellipse having foci at (, ) and (, ) and passing through the origin has eccentricit equal to /7 (B) /7 (C) 5/7 (D) /7 Q.9 Let S(5, ) and S'(, 5) are the foci of an ellipse passing through the origin. The eccentricit of ellipse equals (B) (C) (C) (D) (D)
More than one are correct: Q.0 Consider the ellipse + = where α (0, /). tan α sec α Which of the following quantities would var as α varies? degree of flatness (B) ordinate of the vertex (C) coordinates of the foci (D) length of the latus rectum Q. Extremities of the latera recta of the ellipses + = (a > b) having a given major axis a lies on x = a(a ) (B) x = a (a + ) (C) = a(a + x) (D) = a (a x) Subjective: Q. Consider two concentric circles S : z = and S : z = on the Argand plane. A parabola is drawn through the points where 'S ' meets the real axis and having arbitrar tangent of 'S ' as its directrix. If the locus of the focus of drawn parabola is a conic C then find the area of the quadrilateral formed b the tangents at the ends of the latus-rectum of conic C. DPP-7 Q. Point 'O' is the centre of the ellipse with major axis AB & minor axis CD. Point F is one focus of the ellipse. If OF = 6 & the diameter of the inscribed circle of triangle OCF is, then the product (AB) (CD) is equal to 65 (B) 5 (C) 78 (D) none Q. The -axis is the directrix of the ellipse with eccentricit e = / and the corresponding focus is at (, 0), equation to its auxilar circle is x + 8x + = 0 (B) x + 8x = 0 (C) x + 8x + 9 = 0 (D) x + = x Q. Which one of the following is the common tangent to the ellipses, a + b + = and x b a + a + b =? a = bx + a + b (B) b = ax a + + b (C) a = bx a + + b (D) b = ax + a + b Q. x + = 0 is a common tangent to = x & x + b =. Then the value of b and the other common tangent are given b : b = ; x + + = 0 (B) b = ; x + + = 0 (C) b = ; x + = 0 (D) b = ; x = 0 Q.5 If α & β are the eccentric angles of the extremities of a focal chord of an standard ellipse, then the eccentricit of the ellipse is : cos α + cos β cos( α + β) (B) sin α sin β sin( α β) (C) cos α cos β cos( α β) (D) sin α + sin β sin( α + β) Dpp's on Conic Section (Parabola, Ellipse, Hperbola) [9]
Q.6 An ellipse is inscribed in a circle and a point within the circle is chosen at random. If the probabilit that this point lies outside the ellipse is / then the eccentricit of the ellipse is : (B) 5 (C) 8 9 (D) Q.7 Consider the particle travelling clockwise on the elliptical path + =. The particle leaves the 00 5 orbit at the point ( 8, ) and travels in a straight line tangent to the ellipse. At what point will the particle cross the -axis? 5 5 7 0, (B) 0, (C) (0, 9) (D) 0, Q.8 The Locus of the middle point of chords of an ellipse + = passing through P(0, 5) 6 5 is another ellipse E. The coordinates of the foci of the ellipse E, is 0, and 0, (B) (0, ) and (0, ) 5 5 (C) (0, ) and (0, ) (D) 0, and 0, Paragraph for question nos. 9 to Consider the curve C : 8x + 8 = 0. Tangents TP and TQ are drawn on C at P(5, 6) and Q(5, ). Also normals at P and Q meet at R. Q.9 The coordinates of circumcentre of PQR, is (5, ) (B) (5, ) (C) (5, ) (D) (5, 6) Q.0 The area of quadrilateral TPRQ, is 8 (B) 6 (C) (D) 6 Q. Angle between a pair of tangents drawn at the end points of the chord + 5t = tx + of curve C t R, is 6 (B) (C) (D) More than one are correct: Q. If a number of ellipse be described having the same major axis ut a variable minor axis then the tangents at the ends of their latera recta pass through fixed points which can be (0, a) (B) (0, 0) (C) (0, a) (D) (a, a) Dpp's on Conic Section (Parabola, Ellipse, Hperbola) [0]
DPP-8 Q. The normal at a variable point P on an ellipse + = of eccentricit e meets the axes of the ellipse in Q and R then the locus of the mid-point of QR is a conic with an eccentricit e such that : e is independent of e (B) e = (C) e = e (D) e = /e Q. The area of the rectangle formed b the perpendiculars from the centre of the standard ellipse to the tangent and normal at its point whose eccentric angle is / is : ( ) ab a + b ( a b ) (B) ( a + b )ab (C) ( a b ) ( + b ) ab a a + b (D) ( ) Q. 7MB If P is an point on ellipse with foci S & S and eccentricit is such that ab PS S = α, PS S = β, S PS = γ, then cot α, cot γ, cot β are in A.P. (B) G.P. (C) H.P. (D) NOT A.P., G.P. & H.P. Paragraph for question nos. to 6 Consider the ellipse + = and the parabola = x. The intersect at P and Q in the first and 9 fourth quadrants respectivel. Tangents to the ellipse at P and Q intersect the x-axis at R and tangents to the parabola at P and Q intersect the x-axis at S. Q. The ratio of the areas of the triangles PQS and PQR, is : (B) : (C) : (D) : Q.5 The area of quadrilateral PRQS, is 5 5 5 5 5 (B) (C) (D) Q.6 The equation of circle touching the parabola at upper end of its latus rectum and passing through its vertex, is x + x = 0 (B) x + +x 9 = 0 (C) x + + x = 0 (D) x + 7x + = 0 Paragraph for question nos. 7 to Let the two foci of an ellipse be (, 0) and (, ) and the foot of perpendicular from the focus (, ) upon a tangent to the ellipse be (, 6). Q.7 7MB The foot of perpendicular from the focus (, 0) upon the same tangent to the ellipse is, 5 5 7 (B), 7 (C), (D) (, ) Q.8 The equation of auxiliar circle of the ellipse is x + x 5 = 0 (B) x + x 0 = 0 (C) x + + x + 0 = 0 (D) x + + x + 5 = 0 Dpp's on Conic Section (Parabola, Ellipse, Hperbola) []
Q.9 The length of semi-minor axis of the ellipse is (B) (C) 7 (D) 9 Q.0 The equations of directrices of the ellipse are 7 x + = 0, x 5 = 0 (B) x + = 0, x + + = 0 5 9 (C) x + = 0, x = 0 (D) x + = 0, x + + = 0 Q. The point of contact of the tangent with the ellipse is 0 68, (B), 7 8 8 7 (C), 7 5 5 8 (D), Subjective: x Q. 70MB Find the number of integral values of parameter 'a' for which three chords of the ellipse a = a + (other than its diameter) passing through the point a P a, are bisected b the parabola = ax. DPP-9 Q. Consider the hperbola 9x 6 + 7x 6 = 0. Find the following: (a) centre (b) eccentricit (c) focii (d) equation of directrix (e) length of the latus rectum (f) equation of auxilar circle (g) equation of director circle Q. The area of the quadrilateral with its vertices at the foci of the conics 9x 6 8x + = 0 and 5x + 9 50x 8 + = 0, is 5/6 (B) 8/9 (C) 5/ (D) 6/9 Q. Eccentricit of the hperbola conjugate to the hperbola = is (B) (C) (D) Q. The locus of the point of intersection of the lines x t = 0 & tx + t = 0 (where t is a parameter) is a hperbola whose eccentricit is (B) (C) (D) Q.5 If the eccentricit of the hperbola x sec α = 5 is times the eccentricit of the ellipse x sec α + = 5, then a value of α is : /6 (B) / (C) / (D) / Q.6 The foci of the ellipse + = and the hperbola = coincide. Then the value of b is 6 b 8 5 5 (B) 7 (C) 9 (D) Dpp's on Conic Section (Parabola, Ellipse, Hperbola) []
More than one are correct: Q.7 Which of the following equations in parametric form can represent a hperbola, where 't' is a parameter. x = a b t + & = t t t (B) tx a b + t = 0 & x a + t b = 0 (C) x = e t + e t & = e t e t (D) x 6 = cos t & + = cos t Q.8 Let p and q be non-zero real numbers. Then the equation (px + q + r)(x + 8x ) = 0 represents two straight lines and a circle, when r = 0 and p, q are of the opposite sign. (B) two circles, when p = q and r is of sign opposite to that of p. (C) a hperbola and a circle, when p and q are of opposite sign and r 0. (D) a circle and an ellipse, when p and q are unequal but of same sign and r is of sign opposite to that of p. Match the column: Q.9 Match the properties given in column-i with the corresponding curves given in the column-ii. Column-I Column-II The curve such that product of the distances of an of its tangent (P) Circle from two given points is constant, can be (B) A curve for which the length of the subnormal at an of its point is (Q) Parabola equal to and the curve passes through (, ), can be (C) A curve passes through (, ) and is such that the segment joining (R) Ellipse an point P on the curve and the point of intersection of the normal at P with the x-axis is bisected b the -axis. The curve can be (S) Hperbola (D) A curve passes through (, ) is such that the length of the normal at an of its point is equal to. The curve can be DPP-0 Q. The magnitude of the gradient of the tangent at an extremit of latera recta of the hperbola = is equal to (where e is the eccentricit of the hperbola) be (B) e (C) ab (D) ae Q. The number of possible tangents which can be drawn to the curve x 9 = 6, which are perpendicular to the straight line 5x + 0 = 0 is : zero (B) (C) (D) Q. Locus of the point of intersection of the tangents at the points with eccentric angles φ and hperbola x = is : x = a (B) = b (C) x = ab (D) = ab Dpp's on Conic Section (Parabola, Ellipse, Hperbola) [] φ on the x Q. The equation 9 p + = (p, 9) represents p an ellipse if p is an constant greater than. (B) a hperbola if p is an constant between and 9 (C) a rectangular hperbola if p is an constant greater than 9. (D) no real curve if p is less than 9. Q.5 If = represents famil of hperbolas where α varies then cos α sin α distance between the foci is constant (B) distance between the two directrices is constant (C) distance between the vertices is constant (D) distances between focus and the corresponding directrix is constant
Q.6 Number of common tangent with finite slope to the curves x = c & = ax is : 0 (B) (C) (D) Q.7 Area of the quadrilateral formed with the foci of the hperbola = and = is (a + b ) (B) (a + b ) (C) (a + b ) (D) (a + b ) Q.8 For each positive integer n, consider the point P with abscissa n on the curve x =. If d n represents the shortest distance from the point P to the line = x then Lim(n d ) has the value equal to (B) (C) n n (D) 0 Paragraph for question nos. 9 to The graph of the conic x ( ) = has one tangent line with positive slope that passes through the origin. the point of tangenc being (a, b). Then Q.9 The value of sin a is b 5 (B) (C) (D) 6 Q.0 Length of the latus rectum of the conic is (B) (C) (D) none Q. Eccentricit of the conic is (B) (C) (D) none DPP- Q. If x + i= φ + iψ where i = and φ and ψ are non zero real parameters then φ = constant and ψ = constant, represents two sstems of rectangular hperbola which intersect at an angle of (B) (C) (D) 6 Q. Locus of the feet of the perpendiculars drawn from either foci on a variable tangent to the hperbola 6 9x = is x + = 9 (B) x + = /9 (C) x + =7/ (D) x + = /6 Q. PQ is a double ordinate of the ellipse x + 9 = 9, the normal at P meets the diameter through Q at R, then the locus of the mid point of PR is a circle (B) a parabola (C) an ellipse (D) a hperbola Q. With one focus of the hperbola = as the centre, a circle is drawn which is tangent to the 9 6 hperbola with no part of the circle being outside the hperbola. The radius of the circle is less than (B) (C) (D) none Dpp's on Conic Section (Parabola, Ellipse, Hperbola) []
Q.5 If the tangent and normal at an point of the hperbola =, meets the conjugate axis at Q and R, then the circle described on QR as diameter passes through the vertices (B) focii (C) feet of directrices (D) ends of latera recta Q.6 Let the major axis of a standard ellipse equals the transverse axis of a standard hperbola and their director circles have radius equal to R and R respectivel. If e and e are the eccentricities of the ellipse and hperbola then the correct relation is e e = 6 (B) e e = (C) e e = 6 (D) e e = Q.7 If the normal to the rectangular hperbola x = c at the point 't' meets the curve again at 't ' then t t has the value equal to (B) (C) 0 (D) none Q.8 P is a point on the hperbola =, N is the foot of the perpendicular from P on the transverse axis. The tangent to the hperbola at P meets the transverse axis at T. If O is the centre of the hperbola, the OT. ON is equal to : e (B) a (C) b (D) b /a More than one are correct: d Q.9 Solutions of the differential equation ( x ) + x = ax where a R, is dx a conic which is an ellipse or a hperbola with principal axes parallel to coordinates axes. (B) centre of the conic is (0, a) (C) length of one of the principal axes is. (D) length of one of the principal axes is equal to. Q.0 In which of the following cases maximum number of normals can be drawn from a point P ling in the same plane circle (B) parabola (C) ellipse (D) hperbola Q. If θ is eliminated from the equations a sec θ x tan θ = and b sec θ + tan θ = x (a and b are constant) then the eliminant denotes the equation of the director circle of the hperbola = (B) auxiliar circle of the ellipse + = (C) Director circle of the ellipse + = (D) Director circle of the circle x + = a + b. Dpp's on Conic Section (Parabola, Ellipse, Hperbola) [5]
DPP- Q. If P(x, ), Q(x, ), R(x, ) & S(x, ) are concclic points on the rectangular hperbola = c, the co-ordinates of the orthocentre of the triangle PQR are : (x, ) (B) (x, ) (C) ( x, ) (D) ( x, ) Q. Let F, F are the foci of the hperbola = and F 6 9, F are the foci of its conjugate hperbola. If e H and e C are their eccentricities respectivel then the statement which holds true is Their equations of the asmptotes are different. (B) e H > e C (C) Area of the quadrilateral formed b their foci is 50 sq. units. (D) Their auxillar circles will have the same equation. Q. The chord PQ of the rectangular hperbola x = a meets the axis of x at A ; C is the mid point of PQ & 'O' is the origin. Then the ACO is : equilateral (B) isosceles (C) right angled (D) right isosceles. Q. The asmptote of the hperbola = form with an tangent to the hperbola a triangle whose area is a tan λ in magnitude then its eccentricit is : secλ (B) cosecλ (C) sec λ (D) cosec λ Q.5 Latus rectum of the conic satisfing the differential equation, x d + dx = 0 and passing through the point (, 8) is : (B) 8 (C) 8 (D) 6 Q.6 AB is a double ordinate of the hperbola = such that AOB (where 'O' is the origin) is an equilateral triangle, then the eccentricit e of the hperbola satisfies e > (B) < e < (C) e = (D) e > Q.7 The tangent to the hperbola x = c at the point P intersects the x-axis at T and the -axis at T. The normal to the hperbola at P intersects the x-axis at N and the -axis at N. The areas of the triangles PNT and PN'T' are and ' respectivel, then + is ' equal to (B) depends on t (C) depends on c (D) equal to Q.8 At the point of intersection of the rectangular hperbola x = c and the parabola = ax tangents to the rectangular hperbola and the parabola make an angle θ and φ respectivel with the axis of X, then θ = tan ( tanφ ) (B) φ = tan ( tanθ ) (C) θ = tan ( tanφ ) (D) φ = tan ( tanθ ) Q.9 Locus of the middle points of the parallel chords with gradient m of the rectangular hperbola x = c is + mx = 0 (B) mx = 0 (C) m x = 0 (D) m + x = 0 Q.0 The locus of the foot of the perpendicular from the centre of the hperbola x = c on a variable tangent is (x ) = c x (B) (x + ) = c x (C) (x + ) = x x (D) (x + ) = c x Dpp's on Conic Section (Parabola, Ellipse, Hperbola) [6]
Q. The equation to the chord joining two points (x, ) and (x, ) on the rectangular hperbola x = c is x x + x + + = (B) x x x + = (C) x + + x + x = (D) x + x x = Q. A tangent to the ellipse + = meets its director circle at P and Q. Then the product of the slopes 9 of CP and CQ where 'C' is the origin is 9 (B) 9 (C) 9 (D) Q. The foci of a hperbola coincide with the foci of the ellipse + 5 9 =. Then the equation of the hperbola with eccentricit is = (B) = (C) x + = 0 (D) 9x 5 5 = 0 Paragraph for question nos. to 6 From a point 'P' three normals are drawn to the parabola = x such that two of them make angles with the abscissa axis, the product of whose tangents is. Suppose the locus of the point 'P' is a part of a conic 'C'. Now a circle S = 0 is described on the chord of the conic 'C' as diameter passing through the point (, 0) and with gradient unit. Suppose (a, b) are the coordinates of the centre of this circle. If L and L are the two asmptotes of the hperbola with length of its transverse axis a and conjugate axis b (principal axes of the hperbola along the coordinate axes) then answer the following questions. Q. Locus of P is a circle (B) parabola (C) ellipse (D) hperbola Q.5 Radius of the circle S = 0 is (B) 5 (C) 7 (D) Q.6 The angle α (0, /) between the two asmptotes of the hperbola lies in the interval (0, 5 ) (B) (0, 5 ) (C) (5, 60 ) (D) (60, 75 ) Paragraph for question nos. 7 to 9 A conic C passes through the point (, ) and is such that the segment of an of its tangents at an point contained between the co-ordinate axes is bisected at the point of tangenc. Let S denotes circle described on the foci F and F of the conic C as diameter. Q.7 Vertex of the conic C is (, ), (, ) (B) (, ), (, ) (C) (, ), (, ) (D) (, ), (, ) Q.8 Director circle of the conic is x + = (B) x + = 8 (C) x + = (D) None Q.9 Equation of the circle S is x + = 6 (B) x + = 8 (C) x + = (D) x + = Dpp's on Conic Section (Parabola, Ellipse, Hperbola) [7]
Assertion and Reason: Q.0 Statement-: Diagonals of an parallelogram inscribed in an ellipse alwas intersect at the centre of the ellipse. Statement-: Centre of the ellipse is the onl point at which two chords can bisect each other and ever chord passing through the centre of the ellipse gets bisected at the centre. Statement- is True, Statement- is True ; Statement- is a correct explanation for Statement- (B) Statement- is True, Statement- is True ; Statement- is NOT a correct explanation for Statement- (C) Statement - is True, Statement - is False (D) Statement - is False, Statement - is True Q. Statement-: The points of intersection of the tangents at three distinct points A, B, C on the parabola = x can be collinear. Statement-: If a line L does not intersect the parabola = x, then from ever point of the line two tangents can be drawn to the parabola. Statement- is True, Statement- is True ; Statement- is a correct explanation for Statement- (B) Statement- is True, Statement- is True ; Statement- is NOT a correct explanation for Statement- (C) Statement - is True, Statement - is False (D) Statement - is False, Statement - is True Q. Statement-: The latus rectum is the shortest focal chord in a parabola of length a. because Statement-: As the length of a focal chord of the parabola = ax is a t +, which is minimum t when t =. Statement- is True, Statement- is True ; Statement- is a correct explanation for Statement- (B) Statement- is True, Statement- is True ; Statement- is NOT a correct explanation for Statement- (C) Statement - is True, Statement - is False (D) Statement - is False, Statement - is True Q. Statement-: If P(a, 0) be an point on the axis of parabola, then the chord QPR, satisf (PQ) + =. (PR) a Statement-: There exists a point P on the axis of the parabola = ax (other than vertex), such that (PQ) + = constant for all chord QPR of the parabola. (PR) Statement- is True, Statement- is True ; Statement- is a correct explanation for Statement- (B) Statement- is True, Statement- is True ; Statement- is NOT a correct explanation for Statement- (C) Statement - is True, Statement - is False (D) Statement - is False, Statement - is True Q. Statement-: The quadrilateral formed b the pair of tangents drawn from the point (0, ) to the parabola + x + 5 = 0 and the normals at the point of contact of tangents in a square. Statement-: The angle between tangents drawn from the given point to the parabola is 90. Statement- is True, Statement- is True ; Statement- is a correct explanation for Statement- (B) Statement- is True, Statement- is True ; Statement- is NOT a correct explanation for Statement- (C) Statement - is True, Statement - is False (D) Statement - is False, Statement - is True Dpp's on Conic Section (Parabola, Ellipse, Hperbola) [8]
More than one are correct: Q.5 If the circle x + = a intersects the hperbola x = c in four points P(x, ), Q(x, ), R(x, ), S(x, ), then x + x + x + x = 0 (B) + + + = 0 (C) x x x x = c (D) = c Q.6 The tangent to the hperbola, x = at the point (, 0) when associated with two asmptotes constitutes isosceles triangle (B) an equilateral triangle (C) a triangles whose area is sq. units (D) a right isosceles triangle. Q.7 The locus of the point of intersection of those normals to the parabola x = 8 which are at right angles to each other, is a parabola. Which of the following hold(s) good in respect of the locus? Length of the latus rectum is. (B) Coordinates of focus are 0, (C) Equation of a director circle is = 0 (D) Equation of axis of smmetr = 0. Match the column: Q.8 Column-I Column-II If the chord of contact of tangents from a point P to the (P) Straight line parabola = ax touches the parabola x = b, the locus of P is (B) A variable circle C has the equation (Q) Circle x + (t t + )x (t + t) + t = 0, where t is a parameter. The locus of the centre of the circle is (C) The locus of point of intersection of tangents to an ellipse + = (R) Parabola at two points the sum of whose eccentric angles is constant is (D) An ellipse slides between two perpendicular straight lines. (S) Hperbola Then the locus of its centre is Q.9 Column-I Column-II For an ellipse + 9 = with vertices A and A', tangent drawn at the (P) point P in the first quadrant meets the -axis in Q and the chord A'P meets the -axis in M. If 'O' is the origin then OQ MQ equals to (B) If the product of the perpendicular distances from an point on the (Q) hperbola = of eccentricit e = from its asmptotes is equal to 6, then the length of the transverse axis of the hperbola is (C) The locus of the point of intersection of the lines (R) x t = 0 and tx + t = 0 (where t is a parameter) is a hperbola whose eccentricit is (D) If F & F are the feet of the perpendiculars from the foci S & S (S) 6 of an ellipse + = on the tangent at an point P on the ellipse, 5 then (S F ). (S F ) is equal to Dpp's on Conic Section (Parabola, Ellipse, Hperbola) [9]
ANSWER KEY DPP - Q. B Q. A Q. A Q. B Q.5 A Q.6 D Q.7 B Q.8 C Q.9 D Q.0 C Q. B Q. A Q. B Q. B Q.5 Q.6 b ac b, a a ; b ac b, + ac b a a a ;, = a a a DPP - Q. C Q. D Q. A Q. C Q.5 C Q.6 B Q.7 D Q.8 A Q.9 D Q.0 B Q. A Q. A,B DPP - Q. B Q. B Q. B Q. C Q.5 B Q.6 D Q.7 C Q.8 C Q.9 C Q.0 B Q. C Q. D Q. D Q. C Q.5 A,D Q.6 A,B,C DPP - Q. C Q. A Q. B Q. C Q.5 C Q.6 B Q.7 A Q.8 C Q.9 A Q.0 A Q. C Q. C Q. D DPP -5 Q. B Q. B Q. D Q. D Q.5 C Q.6 A Q.7 A Q.8 B Q.9 C Q.0 C Q. B Q. A Q. A,B,C,D Q. A,B,C Q.5 A,B,C,D Q.6 A,B,D Q.7 Q; (B) R; (C) S; (D) P Q.8 00 DPP -6 Q. D Q. B Q. A Q. C Q.5 C Q.6 A Q.7 B Q.8 C Q.9 C Q.0 A,B,D Q. A,B Q. 006 DPP -7 Q. A Q. A Q. B Q. A Q.5 D Q.6 A Q.7 A Q.8 C Q.9 B Q.0 C Q. D Q. A,C DPP -8 Q. C Q. A Q. A Q. C Q.5 B Q.6 D Q.7 A Q.8 B Q.9 C Q.0 D Q. A Q. 000 DPP -9 Q. (a) (, ); (b) 5/; (c) (, ), ( 9, ); (d) 5x + = 0, 5x + 6 = 0, (e) 9/; (f) (x + ) + ( + ) = 6; (g) (x + ) + ( + ) = 7 Q. B Q. A Q. B Q.5 B Q.6 B Q.7 A,C,D Q.8 A,B,C,D Q.9 R, S; (B) Q; (C) R; (D) P DPP -0 Q. B Q. A Q. B Q. B Q.5 A Q.6 B Q.7 B Q.8 A Q.9 D Q.0 C Q. D DPP - Q. D Q. D Q. C Q. B Q.5 B Q.6 C Q.7 B Q.8 B Q.9 A,B,D Q.0 A Q. C,D DPP - Q. C Q. C Q. B Q. A Q.5 C Q.6 D Q.7 C Q.8 A Q.9 A Q.0 D Q. A Q. B Q. B Q. B Q.5 A Q.6 D Q.7 B Q.8 D Q.9 C Q.0 A Q. D Q. A Q. A Q. D Q.5 A,B,C,D Q.6 B,C Q.7 A,C Q.8 S; (B) R; (C) P; (D) Q Q.9 R; (B) S; (C) P; (D) Q Dpp's on Conic Section (Parabola, Ellipse, Hperbola) [0]