1 is equal to. 1 (B) a. (C) a (B) (D) 4. (C) P lies inside both C & E (D) P lies inside C but outside E. (B) 1 (D) 1

Transcription

1 Single Correct Q. Two mutuall perpendicular tangents of the parabola = a meet the ais in P and P. If S is the focus of the parabola then l a (SP ) is equal to (SP ) l (B) a (C) a Q. ABCD and EFGC are squares and the curve = k passes through the origin D and the points B and FG F. The ratio is BC a (B) (C) Q. Let 'E' be the ellipse 9 + = & 'C' be the circle + = 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 P lies inside C but outside E. Q. The normal at P() on the ellipse cuts the ais of at G and PG is produced to Q so that GQ = PG then the locus of Q is given b a (b a ) b a (C) (a b ) b a (B) (a b ) b (a b ) b Q. Let S be the focus of = and a point P is moving on the curve such that it's abscissa is increasing at the rate of units/sec, then the rate of increase of projection of SP on + = when P is at (, ) is (B) (C) Q.6 Let S (, ) and S' (9, ) be two foci of an ellipse. If the coordinates of the foot of the perpendicular from focus S to a tangent of the ellipse is (, ) then the eccentricit of the ellipse is (B) (C) 7 7 Q.7 The points of contact Q and R of tangent from the point P (, ) on the parabola = are (9, 6) and (, ) (B) (, ) and (, ) (C) (, ) and (9, 6) (9, 6) and (, )

2 Q.8 The eccentricit of the ellipse ( ) + ( ) = 9 is (B) (C) Q.9 A tangent is drawn to the parabola = at the point 'P' whose abscissa lies in the interval [,]. The maimum possible area of the triangle formed b the tangent at 'P', ordinate of the point 'P' and the -ais is equal to 8 (B) 6 (C) Q.0 An ellipse has OB as a semi minor ais 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) Q. From an eternal point P, pair of tangent lines are drawn to the parabola, =. If & are the inclinations of these tangents with the ais of such that, + =, then the locus of P is : + = 0 (B) + = 0 (C) = = 0 Q. The length of the normal (terminated b the major ais) at a point of the ellipse is b b b (r r ) (B) r r (C) a a a r r where r and r are the focal distances of the point. Q. Maimum number of common chords of a parabola and a circle can be equal to (B) (C) 6 8 independent of r, r Q. Through the focus of the parabola = p (p > 0) a line is drawn which intersects the curve at A(, ) and B(, ). The ratio equals (B) (C) some function of p Q. Length of the normal chord of the parabola, =, which makes an angle of 8 (B) 8 (C) with the ais of is: Q.6 There are eactl 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.7 If the lines ( b) = m ( + a) and ( b) = m ( + a) are the tangents to the parabola = a, then m + m = 0 (B) m m = (C) m m = m + m = (C)

3 Q.8 If the normal to a parabola = a at P meets the curve again in Q and if PQ and the normal at Q makes angles and respectivel with the -ais then tan (tan + tan ) has the value equal to 0 (B) (C) Q.9 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 ais of the ellipse is 7 & the area of the triangle PF F is 0, then the distance between the foci is : (B) (C) none Q.0 Let A and B be two points on a parabola = with verte V such that VA is perpendicular to VB and is the angle between the chord VA and the ais of the parabola. The value of tan (B) tan (C) cot cot Q. Minimum distance between the curves = and = is equal to (B) (C) 7 VA is VB Q. Point 'O' is the centre of the ellipse with major ais AB & minor ais 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 6 (B) (C) 78 none Q. The length of a focal chord of the parabola = a at a distance b from the verte is c, then a = bc (B) a = b c (C) ac = b b c = a Q. The straight line joining an point P on the parabola = a to the verte and perpendicular from the focus to the tangent at P, intersect at R, then the equaiton of the locus of R is + a = 0 (B) + a = 0 (C) + a = 0 + a = 0 Q. For a given parabola = a two chords at right angles are drawn through the fied point (, ). If r, r ; r, r are the corresponding pairs of lengths of chords then for all pairs of such lines the value of r is r r r constant (B) equal to a (C) equal to a equal to a Q.6 The -ais is the directri of the ellipse with eccentricit e = / and the corresponding focus is at (, 0), equation to its auilar circle is = 0 (B) + 8 = 0 (C) = 0 + = Q.7 Locus of the feet of the perpendiculars drawn from verte of the parabola = a upon all such chords of the parabola which subtend a right angle at the verte is + a = 0 (B) + a = 0 (C) + + a = a = 0 Q.8 C is the centre of the circle with centre (0, ) and radius unit. P is the parabola = a. The set of values of 'a' for which the meet at a point other than the origin, is a > 0 (B) a 0, (C),,

4 Q.9 TP & TQ are tangents to the parabola, = a at P & Q. If the chord PQ passes through the fied point ( a, b) then the locus of T is : a = b ( b) (B) b = a ( a) (C) b = a ( a) a = b ( b) Q.0 Which one of the following is the common tangent to the ellipses, a b = and b a a b =? a = b + a b (B) b = a a b (C) a = b a b b = a + a b Q. Through the verte O of the parabola, = a two chords OP & OQ are drawn and the circles on OP & OQ as diameters intersect in R. If, & are the angles made with the ais b the tangents at P & Q on the parabola & b OR then the value of, cot + cot = tan (B) tan () (C) 0 cot Q. The normal at a variable point P on an ellipse = of eccentricit e meets the aes 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 e = /e Q. If a normal to a parabola = a make an angle with its ais, then it will cut the curve again at an angle tan ( tan) (B) tan tan (C) cot tan none Q. Tangents are drawn from the points on the line + = 0 to parabola = 8. Then the variable chords of contact pass through a fied point whose coordinates are : (, ) (B) (, ) (C) (, ) (, ) Q. 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 = + 9 and = is (B) (C) Q.6 The area of the quadrilateral with its vertices at the foci of the conics = 0 and = 0, is /6 (B) 8/9 (C) / 6/9 Q.7 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 verte is : a circle (B) a parabola (C) an ellipse (D a hperbola Q.8 The magnitude of the gradient of the tangent at an etremit of latera recta of the hperbola is equal to (where e is the eccentricit of the hperbola) be (B) e (C) ab ae Q.9 The length of the chord of the parabola = which is bisected at the point (, ) is (B) (C)

5 Q.0 If P(, ), Q(, ), R(, ) & S(, ) are concclic points on the rectangular hperbola = c, the co-ordinates of the orthocentre of the triangle PQR are : (, ) (B) (, ) (C) (, ) (, ) Q. + = 0 is a common tangent to = & b =. Then the value of b and the other common tangent are given b : b = ; + + = 0 (B) b = ; + + = 0 (C) b = ; + = 0 b = ; = 0 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 0 sq. units. Their auillar circles will have the same equation. Q. If the tangents & normals at the etremities of a focal chord of a parabola intersect at (, ) and (, ) respectivel, then : = (B) = (C) = = Q. The chord PQ of the rectangular hperbola = a meets the ais of at A ; C is the mid point of PQ & 'O' is the origin. Then the ACO is : equilateral (B) isosceles (C) right angled right isosceles. Q. The locus of point of intersection of tangents to the parabolas = ( + ) & = 8( + ) which are perpendicular to each other is : + 7 = 0 (B) = (C) + = 0 = Q.6 Eccentricit of the hperbola conjugate to the hperbola is (B) (C) Q.7 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 : a b ab a b a b a b (B) (C) a b a b ab ab a b a b ab Q.8 The asmptote of the hperbola = form with an tangent to the hperbola a triangle whose area is a tan in magnitude then its eccentricit is : sec (B) cosec (C) sec cosec Q.9 If two normals to a parabola = a intersect at right angles then the chord joining their feet passes through a fied point whose co-ordinates are : ( a, 0) (B) (a, 0) (C) (a, 0) none

6 Q.0 A conic passes through the point (, ) and is such that the segment of an of its tangents at an point contained between the co-ordinate aes is bisected at the point of tangenc. Then the foci of the conic are, 0 &, 0 (B), &, (C) (, ) & (, ), &, Q. The equation of a straight line passing through the point (, 6) and cutting the curve = orthogonall is + 8 =0 (B) + 9 = 0 (C) 6 = 0 none Q. The number of possible tangents which can be drawn to the curve 9 = 6, which are perpendicular to the straight line + 0 = 0 is : zero (B) (C) Q. A tangent to the parabola + a = 0 cuts the parabola = b at A and B. The locus of the mid point of AB has the equation : (a + b) = b (B) (b + a) = b (C) (a + b) = b (b + ) = a Q. Locus of the point of intersection of the tangents at the points with eccentric angles and on the hperbola = is : = a (B) = b (C) = ab = ab Q. Imagine that ou have two thumbtacks placed at two points, A and B. If the ends of a fied 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 maimise the area surrounded b the ellipse with a fied length of string occurs when I the two points A and B have the maimum 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 IV Q.6 The equation 9 p + p = (p, 9) represents 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. no real curve if p is less than 9. Q.7 If + i= i where i = and and are non zero real parameters then = constant and = constant, represents two sstems of rectangular hperbola which intersect at an angle of (B) (C) 6 Q.8 PQ is a normal chord of the parabola = a at P, A being the verte of the parabola. Through P a line is drawn parallel to AQ meeting the ais 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 equal to the distance of the point P from the directri.

7 Q.9 Locus of the feet of the perpendiculars drawn from either foci on a variable tangent to the hperbola 6 9 = is + = 9 (B) + = /9 (C) + =7/ + = /6 Q.60 PQ is a double ordinate of the ellipse + 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 a hperbola Q.6 If & are the eccentric angles of the etremities of a focal chord of an standard ellipse, then the eccentricit of the ellipse is : cos cos cos( ) (B) sin sin sin( ) (C) cos cos cos( ) sin sin sin( ) Q.6 Latus rectum of the conic satisfing the differential equation, d + d = 0 and passing through the point (, 8) is : (B) 8 (C) 8 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 = e > Q.6 Length of the focal chord of the parabola = a at a distance p from the verte is : a p (B) a (C) a p p p a Q.6 The locus of the point of intersection of the lines t = 0 & t + t = 0 (where t is a parameter) is a hperbola whose eccentricit is (B) (C) Q.66 If = represents famil of hperbolas where varies then cos sin distance between the foci is constant (B) distance between the two directrices is constant (C) distance between the vertices is constant distances between focus and the corresponding directri is constant Q.67 The triangle PQR of area 'A' is inscribed in the parabola = a such that the verte P lies at the verte 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 A a Q.68 The tangent to the hperbola = c at the point P intersects the -ais at T and the -ais at T. The normal to the hperbola at P intersects the -ais at N and the -ais 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 equal to

8 Q.69 Number of common tangent with finite slope to the curves = c & = a is : 0 (B) (C) Q.70 The equation of the circle drawn with the focus of the parabola ( ) 8 = 0 as its centre and touching the parabola at its verte is : + = 0 (B) + + = 0 (C) + = = 0 Q.7 If the eccentricit of the hperbola sec = is times the eccentricit of the ellipse sec + =, then a value of is : /6 (B) / (C) / / Q.7 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) / none Q.7 If the tangent and normal at an point of the hperbola, meets the conjugate ais at Q and R, then the circle described on QR as diameter passes through the vertices (B) focii (C) feet of directrices ends of latera recta Q.7 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) (C) 8 9 Q.7 At the point of intersection of the rectangular hperbola = c and the parabola = a tangents to the rectangular hperbola and the parabola make an angle and respectivel with the ais of X, then = tan ( tan) (B) = tan ( tan) (C) = tan ( tan) = tan ( tan) Q.76 If = 9 is the chord of contact of the hperbola = 9, then the equation of the corresponding pair of tangents, is : = 0 (B) = 0 (C) = = 0 Q.77 The circle drawn on the latus rectum of the parabola + = ( + ) as diameter cuts the ais of the parabola at the points : 9,,, 9 (B),,, (C),, (0, 0) 7,, 9, Q.78 Area of the quadrilateral formed with the foci of the hperbola and is (a + b ) (B) (a + b ) (C) (a + b ) (a + b )

9 Q.79 The foci of the ellipse and the hperbola coincide. Then the value of b 6 is b 8 (B) 7 (C) 9 Q.80 The tangent and normal at P(t), for all real positive t, to the parabola = a meet the ais 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 tan t Q.8 Locus of the middle points of the parallel chords with gradient m of the rectangular hperbola = c is + m = 0 (B) m = 0 (C) m = 0 m + = 0 Q.8 The point of intersection of the tangents at the point P on the ellipse = and its corresponding point Q on the auiliar circle meet on the line : = a/e (B) = 0 (C) = 0 None Q.8 Let the major ais of a standard ellipse equals the transverse ais 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 e e = Q.8 The locus of the foot of the perpendicular from the centre of the hperbola = c on a variable tangent is ( ) = c (B) ( + ) = c (C) ( + ) = ( + ) = c Q.8 For each positive integer n, consider the point P with abscissa n on the curve =. If d n represents the shortest distance from the point P to the line = then Lim(n d ) has the value equal to (B) (C) 0 Q.86 The roots of the equation m m + = 0 are the slopes of the two tangents to the parabola =. The tangent intersect at the point n, (B), (C), point of intersection can not be found as the tangents are not real Q.87 If the normal to the rectangular hperbola = c at the point 't' meets the curve again at 't ' then t t has the value equal to (B) (C) 0 none Q.88 A circle with radius unit has its centre on the positive -ais. If this circle touches the parabola = tangentiall at the points P and Q then the sum of the ordinates of P and Q, is (B) 8 (C) Q.89 P is a point on the hperbola =, N is the foot of the perpendicular from P on the transverse ais. The tangent to the hperbola at P meets the transverse ais at T. If O is the centre of the hperbola, the OT. ON is equal to : e (B) a (C) b b /a n

10 Q.90 An ellipse having foci at (, ) and (, ) and passing through the origin has eccentricit equal to (B) (C) Q.9 The equation to the chord joining two points (, ) and (, ) on the rectangular hperbola = c is + = (B) + = (C) + = + = Paragraph for question nos. 9 to 9 Consider the circles S : = 0; S : + + = 0 and a variable circle S : + + g + f + c = 0 Q.9 Number of common tangents to S and S is (B) (C) Q.9 Length of a transverse common tangent to S and S is 6 (B) (C) Q.9 If the variable circle S = 0 with centre C moves in such a wa that it is alwas touching eternall the circles S = 0 and S = 0 then the locus of the centre C of the variable circle is a circle (B) a parabola (C) an ellipse a hperbola Paragraph for question nos. 9 to 97 From a point 'P' three normals are drawn to the parabola = such that two of them make angles with the abscissa ais, 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 ais a and conjugate ais b (principal aes of the hperbola along the coordinate aes) then answer the following questions. Q.9 Locus of P is a circle (B) parabola (C) ellipse hperbola Q.96 Radius of the circle S = 0 is (B) (C) 7 Q.97 The angle (0, /) between the two asmptotes of the hperbola lies in the interval (0, ) (B) (0, ) (C) (, 60 ) (60, 7 )

11 Paragraph for question nos. 98 to 00 The graph of the conic ( ) = has one tangent line with positive slope that passes through the origin. the point of tangenc being (a, b). Then Q.98 The value of sin a is b (B) 6 Q.99 Length of the latus rectum of the conic is (C) (B) (C) none Q.00 Eccentricit of the conic is (B) (C) none More than one are correct: Q.0 Consider a circle with its centre ling on the focus of the parabola, = p such that it touches the directri of the parabola. Then a point of intersection of the circle & the parabola is : p p, p (B), p (C) p p, p p, Q.0 If the circle + = a intersects the hperbola = c in four points P(, ), Q(, ), R(, ), S(, ), then = 0 (B) = 0 (C) = c = c Q.0 Let = a be a parabola and + + b = 0 be a circle. If parabola and circle touch each other eternall then : a > 0, b > 0 (B) a > 0, b < 0 (C) a < 0, b > 0 a < 0, b < 0 Q.0 The tangent to the hperbola, = at the point, 0 when associated with two asmptotes constitutes : isosceles triangle (B) an equilateral triangle (C) a triangles whose area is sq. units a right isosceles triangle. Q.0 Consider the ellipse = where (0, /). tan sec Which of the following quantities would var as varies? degree of flatness (B) ordinate of the verte (C) coordinates of the foci length of the latus rectum Q.06 Which of the following equations in parametric form can represent a hperbola, where 't' is a parameter. = a b t & = t t t (B) t a b + t = 0 & a + t b = 0 (C) = e t + e t & = e t e t 6 = cos t & + = cos t

12 Q.07 Let P, Q and R are three co-normal points on the parabola = a. 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 ais of the parabola circle circumscribing the triangle PQR passes through the verte of the parabola Q.08 Equations of a common tangent to the two hperbolas a = + a = & b a b (B) = a b (C) = + a b a b = is : b Q.09 Variable chords of the parabola = a subtend a right angle at the verte. Then : locus of the feet of the perpendiculars from the verte on these chords is a circle (B) locus of the middle points of the chords is a parabola (C) variable chords passes through a fied point on the ais of the parabola none of these Q.0 The differential equation d d = represents a famil of hperbolas (ecept when it represents a pair of lines) with eccentricit : (B) (C) Q. Etremities of the latera recta of the ellipses (a > b) having a given major ais a lies on = a(a ) (B) = a (a + ) (C) = a(a + ) = a (a ) Q. Circles are drawn on chords of the rectangular hperbola = c parallel to the line = as diameters. All such circles pass through two fied points whose co-ordinates are : (c, c) (B) (c, c) (C) ( c, c) ( c, c) Q. The straight line + = touches the parabola : + = 0 (B) + = 0 (C) + = 0 + = 0 d Q. Solutions of the differential equation ( ) + = a where a R, is d a conic which is an ellipse or a hperbola with principal aes parallel to coordinates aes. (B) centre of the conic is (0, a) (C) length of one of the principal aes is. length of one of the principal aes is equal to. Q. Through a point P (, 0), tangents PQ and PR are drawn to the parabola = 8. 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 circumcentre Q.6 In which of the following cases maimum number of normals can be drawn from a point P ling in the same plane circle (B) parabola (C) ellipse hperbola

13 Q.7 If a number of ellipse be described having the same major ais ut a variable minor ais then the tangents at the ends of their latera recta pass through fied points which can be (0, a) (B) (0, 0) (C) (0, a) (a, a) Q.8 If is eliminated from the equations a sec tan = and b sec + tan = (a and b are constant) then the eliminant denotes the equation of the director circle of the hperbola (B) auiliar circle of the ellipse (C) Director circle of the ellipse Director circle of the circle + = a 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 -ais is bisected b the -ais. The curve can be (S) Hperbola A curve passes through (, ) is such that the length of the normal at an of its point is equal to. The curve can be Q.0 Consider the parabola = Column-I Column-II Tangent and normal at the etremities of the latus rectum intersect (P) (0, 0) the ais 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 fied point K on (Q) (, 0) the ais, 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 point K are (R) (6, 0) (C) All variable chords of the parabola subtending a right angle at the origin are concurrent at the point AB and CD are the chords of the parabola which intersect at a point (S) (, 0) E on the ais. The radical ais of the two circles described on AB and CD as diameter alwas passes through b.

14 Q. Column-I Column-II If the chord of contact of tangents from a point P to the (P) Straight line parabola = a touches the parabola = b, the locus of P is (B) A variable circle C has the equation (Q) Circle + (t t + ) (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 An ellipse slides between two perpendicular straight lines. (S) Hperbola Then the locus of its centre is Q. 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 -ais in Q and the chord A'P meets the -ais 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 ais of the hperbola is (C) The locus of the point of intersection of the lines (R) t = 0 and t + t = 0 (where t is a parameter) is a hperbola whose eccentricit is 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, then (S F ). (S F ) is equal to