QUESTION BANK ON. CONIC SECTION (Parabola, Ellipse & Hyperbola)

Transcription

1 QUESTION BANK ON CONIC SECTION (Parabola, Ellipse & Hyperbola)

2 Question bank on Parabola, Ellipse & Hyperbola Select the correct alternative : (Only one is correct) Q. Two mutually perpendicular tangents of the parabola y = 4ax meet the axis in P and P. If S is the focus of the parabola then l (A) a 4 (SP ) is equal to (SP ) l (B) a (C) a Q. Which one of the following equations represented parametrically, represents equation to a parabolic profile? (D) 4a (A) x = cos t ; y = 4 sint (B) x = cos t ; y = 4 cos t (C) x = tan t ; y = sec t (D) x = sin t ; y = sin t + cos t Q. The magnitude of the gradient of the tangent at an extremity of latera recta of the hyperbola a b is equal to (where e is the eccentricity of the hyperbola) (A) be (B) e (C) ab (D) ae Q.4 Let 'E' be the ellipse x 9 + y 4 = & 'C' be the circle x + y = 9. Let P & Q be the points (, ) and (, ) respectively. Then : (A) 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.5 Let S be the focus of y = 4x and a point P is moving on the curve such that it's abscissa is increasing at the rate of 4 units/sec, then the rate of increase of projection of SP on x + y = when P is at (4, 4) is (A) (B) (C) (D) Q.6 Eccentricity of the hyperbola conjugate to the hyperbola is 4 4 (A) (B) (C) (D) Q.7 The points of contact Q and R of tangent from the point P (, ) on the parabola y = 4x are (A) (9, 6) and (, ) (B) (, ) and (4, 4) (C) (4, 4) and (9, 6) (D) (9, 6) and ( 4, ) []

3 Q.8 The eccentricity of the ellipse (x ) + (y 4) = 9 y is (A) (B) Q.9 The asymptote of the hyperbola = form with any tangent to the hyperbola a triangle whose a b area is a tan in magnitude then its eccentricity is : (A) sec (B) cosec (C) sec (D) cosec Q.0 A tangent is drawn to the parabola y = 4x at the point 'P' whose abscissa lies in the interval [,4]. The maximum possible area of the triangle formed by the tangent at 'P', ordinate of the point 'P' and the x-axis is equal to (A) 8 (B) 6 (C) 4 (D) Q. From an external point P, pair of tangent lines are drawn to the parabola, y = 4x. If & are the inclinations of these tangents with the axis of x such that, + =, then the locus of P is : 4 (A) x y + = 0 (B) x + y = 0 (C) x y = 0 (D) x + y + = 0 Q. The equation x 9 p + 4 p y (C) = (p 4, 9) represents (A) an ellipse if p is any constant greater than 4. (B) a hyperbola if p is any constant between 4 and 9. (C) a rectangular hyperbola if p is any constant greater than 9. (D) no real curve if p is less than 9. Q. For an ellipse with vertices A and A', tangent drawn at the point P in the first quadrant meets 9 4 the y-axis in Q and the chord A'P meets the y-axis in M. If 'O' is the origin then OQ MQ equals to (A) 9 (B) (C) 4 (D) 5 Q.4 Length of the normal chord of the parabola, y = 4x, which makes an angle of 4 with the axis of x is: (A) 8 (B) 8 (C) 4 (D) 4 Q.5 An ellipse and a hyperbola have the same centre origin, the same foci and the minor-axis of the one is the same as the conjugate axis of the other. If e, e be their eccentricities respectively, then e e equals (A) (B) (C) (D) 4 Q.6 The coordiantes of the ends of a focal chord of a parabola y = 4ax are (x, y ) and (x, y ) then x x + y y has the value equal to (A) a (B) a (C) a (D) 4a (D) []

4 Q.7 The line, lx + my + n = 0 will cut the ellipse x + y = in points whose eccentric angles differ by a b / if : (A) a l + b n = m (B) a m + b l = n (C) a l + b m = n (D) a n + b m = l Q.8 Locus of the feet of the perpendiculars drawn from either foci on a variable tangent to the hyperbola 6y 9x = is (A) x + y = 9 (B) x + y = /9 (C) x + y =7/44 (D) x + y = /6 Q.9 If the normal to a parabola y = 4ax at P meets the curve again in Q and if PQ and the normal at Q makes angles and respectively with the x-axis then tan (tan + tan ) has the value equal to (A) 0 (B) (C) (D) Q.0 If the normal to the parabola y = 4ax at the point with parameter t, cuts the parabola again at the point with parameter t, then (A) < t < 8 (B) < t < 4 (C) t > 4 (D) t > 8 Q. The locus of the point of instruction of the lines x y 4 t = 0 & tx + ty 4 = 0 (where t is a parameter) is a hyperbola whose eccentricity is 4 (A) (B) (C) (D) Q. The equation to the locus of the middle point of the portion of the tangent to the ellipse x 6 + y 9 = included between the co-ordinate axes is the curve : (A) 9x + 6y = 4 x y (B) 6x + 9y = 4 x y (C) x + 4y = 4 x y (D) 9x + 6y = x y Q. A parabola y = ax + bx + c crosses the xaxis at (, 0) (, 0) both to the right of the origin. A circle also passes through these two points. The length of a tangent from the origin to the circle is : (A) bc a (B) ac (C) b a (D) c a Q.4 Two parabolas have the same focus. If their directrices are the xaxis & the yaxis respectively, then the slope of their common chord is : (A) ± (B) 4/ (C) /4 (D) none Q.5 The locus of a point in the Argand plane that moves satisfying the equation, z + i z i = (A) is a circle with radius & centre at z = / (B) is an ellipse with its foci at i and + i and major axis = (C) is a hyperbola with its foci at i and + i and its transverse axis = (D) is none of the above. [4]

5 Q.6 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 4 points. Let 'P' be any 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 : (A) (B) (C) (D) none Q.7 The straight line joining any point P on the parabola y = 4ax 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 (A) x + y ax = 0 (B) x + y ax = 0 (C) x + y ay = 0 (D) x + y ay = 0 Q.8 A normal chord of the parabola y = 4x subtending a right angle at the vertex makes an acute angle with the x-axis, then equals to (A) arc tan (B) arc sec (C) arc cot (D) none Q.9 If the eccentricity of the hyperbola x y sec = 5 is times the eccentricity of the ellipse x sec + y = 5, then a value of is : (A) /6 (B) /4 (C) / (D) / Q.0 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 (A) 65 (B) 5 (C) 78 (D) none Q. Locus of the feet of the perpendiculars drawn from vertex of the parabola y = 4ax upon all such chords of the parabola which subtend a right angle at the vertex is (A) x + y 4ax = 0 (B) x + y ax = 0 (C) x + y + ax = 0 (D) x + y + 4ax = 0 Q. For all real values of m, the straight line y = mx + 9m 4 is a tangent to the curve : (A) 9x + 4y = 6 (B) 4x + 9y = 6 (C) 9x 4y = 6 (D) 4x 9y = 6 Q. C is the centre of the circle with centre (0, ) and radius unity. P is the parabola y = ax. The set of values of 'a' for which they meet at a point other than the origin, is (A) a > 0 (B) a 0, (C), (D), 4 Q.4 A tangent having slope of 4 x to the ellipse 8 + y = intersects the major & minor axes in points A & B respectively. If C is the centre of the ellipse then the area of the triangle ABC is : (A) sq. units (B) 4 sq. units (C) 6 sq. units (D) 48 sq. units Q.5 The foci of the ellipse and the hyperbola coincide. Then the value of b 6 is b (A) 5 (B) 7 (C) 9 (D) 4 [5]

6 Q.6 TP & TQ are tangents to the parabola, y = 4ax at P & Q. If the chord PQ passes through the fixed point (a, b) then the locus of T is : (A) ay = b (x b) (B) bx = a (y a) (C) by = a (x a) (D) ax = b (y b) Q.7 Through the vertex O of the parabola, y = 4ax two chords OP & OQ are drawn and the circles on OP & OQ as diameters intersect in R. If, & are the angles made with the axis by the tangents at P & Q on the parabola & by OR then the value of, cot + cot = (A) tan (B) tan () (C) 0 (D) cot Q.8 Locus of the middle points of the parallel chords with gradient m of the rectangular hyperbola xy = c is (A) y + mx = 0 (B) y mx = 0 (C) my x = 0 (D) my + x = 0 Q.9 If the chord through the point whose eccentric angles are & on the ellipse, (x /a ) + (y /b ) = passes through the focus, then the value of ( +e) tan(/) tan(/) is (A) e + (B) e (C) e (D) 0 Q.40 The given circle x + y + px = 0, p R touches the parabola y = 4x externally, then (A) p < 0 (B) p > 0 (C) 0 < p < (D) p < Q.4 The locus of the foot of the perpendicular from the centre of the hyperbola xy = c on a variable tangent is : (A) (x y ) = 4c xy (B) (x + y ) = c xy (C) (x + y ) = 4x xy (D) (x + y ) = 4c xy Q.4 The tangent at P to a parabola y = 4ax meets the directrix at U and the latus rectum at V then SUV (where S is the focus) : (A) must be a right triangle (B) must be an equilateral triangle (C) must be an isosceles triangle (D) must be a right isosceles triangle. Q.4 Given the base of a triangle and sum of its sides then the locus of the centre of its incircle is (A) straight line (B) circle (C) ellipse (D) hyperbola Q.44 P is a point on the hyperbola x y =, N is the foot of the perpendicular from P on the transverse a b axis. The tangent to the hyperbola at P meets the transverse axis at T. If O is the centre of the hyperbola, the OT. ON is equal to : (A) e (B) a (C) b (D)b /a Q.45 Two parabolas y = 4a(x - l ) and x = 4a (y l ) always touch one another, the quantities l and l are both variable. Locus of their point of contact has the equation (A) xy = a (B) xy = a (C) xy = 4a (D) none Q.46 If a normal to a parabola y = 4ax make an angle with its axis, then it will cut the curve again at an angle (A) tan ( tan) (B) tan tan (C) cot tan (D) none [6]

7 Q.47 If PN is the perpendicular from a point on a rectangular hyperbola x y = a on any of its asymptotes, then the locus of the mid point of PN is : (A) a circle (B) a parabola (C) an ellipse (D) a hyperbola x Q.48 Which one of the following is the common tangent to the ellipses, a b y = & x b a a y b =? 4 4 (A) ay = bx + a a b b (B) by = ax a a b b (C) ay = bx 4 4 a a b b (D) by = ax + a a b b Q.49 The vertex of a parabola is (,) and the co-ordinates of its two extrimities of the latus rectum are (,0) and (6,0). The equation of the parabola is (A) y 4y + 8x = 0 (B) x + 4x 8y = 0 (C) x 4x + 8y = 0 (D) x 8y 4x + 0 = 0 Q.50 The equation to the chord joining two points (x, y ) and (x, y ) on the rectangular hyperbola xy = c is (A) + = (B) + x y x y = (C) + = (D) + y y x x y y x x = Q.5 The length of the chord of the parabola y = x which is bisected at the point (, ) is (A) (B) 4 (C) (D) 5 Q.5 The normal at a variable point P on an ellipse x y = of eccentricity e meets the axes of the ellipse a b in Q and R then the locus of the mid-point of QR is a conic with an eccentricity e such that : (A) e is independent of e (B) e = (C) e = e (D) e = /e Q.5 If the tangents & normals at the extremities of a focal chord of a parabola intersect at (x, y ) and (x, y ) respectively, then : (A) x = x (B) x = y (C) y = y (D) x = y Q.54 If P(x, y ), Q(x, y ), R(x, y ) & S(x 4, y 4 ) are 4 concyclic points on the rectangular hyperbola = c, the co-ordinates of the orthocentre of the triangle PQR are : (A) (x 4, y 4 ) (B) (x 4, y 4 ) (C) ( x 4, y 4 ) (D) ( x 4, y 4 ) Q.55 If the chord of contact of tangents from a point P to the parabola y = 4ax touches the parabola x = 4by, the locus of P is : (A) circle (B) parabola (C) ellipse (D) hyperbola [7]

8 Q.56 An ellipse is drawn with major and minor axes of lengths 0 and 8 respectively. Using one focus as centre, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. The radius of the circle is (A) (B) (C) (D) 5 Q.57 The latus rectum of a parabola whose focal chord PSQ is such that SP = and SQ = is given by (A) 4/5 (B) /5 (C) 6/5 (D) none of these Q.58 The chord PQ of the rectangular hyperbola xy = a meets the axis of x at A ; C is the mid point of PQ & 'O' is the origin. Then the ACO is : (A) equilateral (B) isosceles (C) right angled (D) right isosceles. Q.59 The circle x + y = 5 meets the parabola y = 4x at P & Q. Then the length PQ is equal to (A) (B) (C) 4 (D) none Q.60 A common tangent to 9x + 6y = 44 ; y x + 4 = 0 & x + y x + = 0 is (A) y = (B) x = 4 (C) x = 4 (D) y = Q.6 A conic passes through the point (, 4) and is such that the segment of any of its tangents at any point contained between the co-ordinate axes is bisected at the point of tangency. Then the foci of the conic are (A) 0 0,, &, (B), & (C) (4, 4) & ( 4, 4) (D) 4, 4 & 4, 4 Q.6 If two normals to a parabola y = 4ax intersect at right angles then the chord joining their feet passes through a fixed point whose co-ordinates are (A) ( a, 0) (B) (a, 0) (C) (a, 0) (D) none Q.6 The equation of a straight line passing through the point (, 6) and cutting the curve y = x orthogonally is (A) 4x + y 8 =0 (B) x + y 9 = 0 (C) 4x y 6 = 0 (D) none Q.64 Latus rectum of the conic satisfying the differential equation, xdy + ydx = 0 and passing through the point (, 8) is (A) 4 (B) 8 (C) 8 (D) 6 Q.65 The area of the rectangle formed by the perpendiculars from the centre of the standard ellipse to the tangent and normal at its point whose eccentric angle is /4 is a b ab a b a b a b (A) (B) (C) (D) a b a b ab ab a b a b ab [8]

9 Q.66 PQ is a normal chord of the parabola y = 4ax at P, A being the vertex of the parabola. Through P a line is drawn parallel to AQ meeting the xaxis in R. Then the length of AR is : (A) 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.67 If the normal to the rectangular hyperbola xy = c at the point 't' meets the curve again at 't ' then t t has the value equal to (A) (B) (C) 0 (D) none Q.68 Locus of the point of intersection of the perpendicular tangents of the curve y + 4y 6x = 0 is : (A) x = 0 (B) x + = 0 (C) y + = 0 (D) x + 5 = 0 Q.69 If tan. tan = a then the chord joining two points b & on the ellipse x a a right angle at : (A) focus (B) centre (C) end of the major axis (D) end of the minor axis y b = will subtend Q.70 With one focus of the hyperbola as the centre, a circle is drawn which is tangent to the 9 6 hyperbola with no part of the circle being outside the hyperbola. The radius of the circle is (A) less than (B) (C) (D) none Q.7 Length of the focal chord of the parabola y = 4ax at a distance p from the vertex is : (A) a p (B) a (C) 4 a p p (D) p a Q.7 The locus of a point such that two tangents drawn from it to the parabola y = 4ax are such that the slope of one is double the other is : (A) y = 9 ax (B) y = 9 4 ax (C) y = 9 ax (D) x = 4 ay Q.7 AB is a double ordinate of the hyperbola such that AOB (where 'O' is the origin) is an a b equilateral triangle, then the eccentricity e of the hyperbola satisfies (A) e > (B) < e < (C) e = (D) e > Q.74 An ellipse is inscribed in a circle and a point within the circle is chosen at random. If the probability that this point lies outside the ellipse is / then the eccentricity of the ellipse is : (A) (B) 5 (C) 8 9 (D) [9]

10 Q.75 The triangle PQR of area 'A' is inscribed in the parabola y = 4ax 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 a (B) A a (C) A a (D) 4A a Q.76 If the product of the perpendicular distances from any point on the hyperbola of eccentricity a b e = from its asymptotes is equal to 6, then the length of the transverse axis of the hyperbola is (A) (B) 6 (C) 8 (D) Q.77 The point(s) on the parabola y = 4x which are closest to the circle, x + y 4y + 8 = 0 is/are : (A) (0, 0) (B), (C) (4, 4) (D) none Q.78 A point P moves such that the sum of the angles which the three normals makes with the axis drawn from P on the standard parabola, is constant. Then the locus of P is : (A) a straight line (B) a circle (C) a parabola (D) a line pair Q.79 If x + iy = i where i = and and are non zero real parameters then = constant and = constant, represents two systems of rectangular hyperbola which intersect at an angle of (A) (B) (C) (D) 6 4 Q.80 Three normals drawn from any point to the parabola y = 4ax cut the line x = a in points whose ordinates are in arithmetical progression. Then the tangents of the angles which the normals make the axis of the parabola are in : (A) A.P. (B) G.P. (C) H.P. (D) none Q.8 A circle is described whose centre is the vertex and whose diameter is three-quarters of the latus rectum of the parabola y = 4ax. If PQ is the common chord of the circle and the parabola and L L is the latus rectum, then the area of the trapezium PL L Q is : (A) a (B) a (C) 4a (D) a Q.8 The tangent to the hyperbola xy = c at the point P intersects the x-axis at T and the y-axis at T. The normal to the hyperbola at P intersects the x-axis at N and the y-axis at N. The areas of the triangles PNT and PN'T' are and ' respectively, then is ' (A) equal to (B) depends on t (C) depends on c (D) equal to Q.8 If y = x is a tangent to the parabola y = 4a x (A) (B) (C) 4, then ' a ' is equal to : (D) 4 [0]

11 Q.84 An ellipse having foci at (, ) and ( 4, 4) and passing through the origin has eccentricity equal to 5 (A) (B) (C) (D) Q.85 The ellipse 4x + 9y = 6 and the hyperbola 4x y = 4 have the same foci and they intersect at right angles then the equation of the circle through the points of intersection of two conics is (A) x + y = 5 (B) 5 (x + y ) x 4y = 0 (C) 5 (x + y ) + x + 4y = 0 (D) x + y = 5 Q.86 Tangents are drawn from the point (, ) on the parabola y = 4x. The length, these tangents will intercept on the line x = is : (A) 6 (B) 6 (C) 6 (D) none of these Q.87 The curve describes parametrically by x = t t +, y = t + t + represents (A) straight line (B) pair of straight lines (C) circle (D) parabola Q.88 At the point of intersection of the rectangular hyperbola xy = c and the parabola y = 4ax tangents to the rectangular hyperbola and the parabola make an angle and respectively with the axis of X, then (A) = tan ( tan) (B) = tan ( tan) (C) = tan ( tan) (D) = tan ( tan) Q.89 The tangent and normal at P(t), for all real positive t, to the parabola y = 4ax meet the axis of the parabola in T and G respectively, 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 (A) cot t (B) cot t (C) tan t (D) tan t Q.90 Area of the quadrilateral formed with the foci of the hyperbola and is a b a b (A) 4(a + b ) (B) (a + b ) (C) (a + b ) (D) (a + b ) Q.9 A bar of length 0 units moves with its ends on two fixed straight lines at right angles. A point P marked on the bar at a distance of 8 units from one end describes a conic whose eccentricity is (A) 9 5 (B) (C) 9 4 Q.9 In a square matrix A of order, a i i = m i + i where i =,, and m i 's are the slopes (in increasing order of their absolute value) of the normals concurrent at the point (9, 6) to the parabola y = 4x. Rest all other entries of the matrix are one. The value of det. (A) is equal to (A) 7 (B) 6 (C) 4 (D) 9 x Q.9 An equation for the line that passes through (0, ) and is perpendicular to y = is 4 (A) 4x + y = 9 (B) x + y = 9 (C) x + y = 9 (D) x + y = 8 (D) 5 []

12 Direction for Q.94 to Q.97. (4 questions together) A quadratic polynomial y = f (x) with absolute term neither touches nor intersects the abscissa axis and is symmetric about the line x =. The coefficient of the leading term of the polynomial is unity. A point A(x, y ) with abscissa x = and a point B(x, y ) with ordinate y = are given in a cartisian rectangular system of co-ordinates OXY in the first quadrant on the curve y = f (x) where 'O' is the origin. Now answer the following questions: Q.94 Vertex of the quadratic polynomial is (A) (, ) (B) (, ) (C) (, ) (D) none Q.95 The scalar product of the vectors OA and OB is (A) 8 (B) 6 (C) (D) Q.96 The area bounded by the curve y = f(x) and a line y = is (A) 4/ (B) 5/ (C) 7/ (D) 8/ Q.97 The graph of y = f(x) represents a parabola whose focus has the co-ordinates (A) (, 7/4) (B) (, 5/4) (C) (, 5/) (D) (, 9/4) Direction for Q.98 to Q.66. ( questions together) The graph of the conic x (y ) = has one tangent line with positive slope that passes through the origin. the point of tangency being (a, b). Then Q.98 The value of sin a is b 5 (A) (B) 6 Q.99 Length of the latus rectum of the conic is (C) (D) 4 (A) (B) (C) (D) none Q.00 Eccentricity of the conic is (A) 4 (B) (C) (D) none Select the correct alternatives : (More than one are correct) Q.0 Consider a circle with its centre lying on the focus of the parabola, y = px such that it touches the directrix of the parabola. Then a point of intersection of the circle & the parabola is : (A) p p, p (B), p (C) p p, (D) p p, Q.0 Identify the statements which are True. (A) the equation of the director circle of the ellipse, 5x + 9y = 45 is x + y = 4. (B) the sum of the focal distances of the point (0, 6) on the ellipse x 5 + y = is 0. 6 (C) the point of intersection of any tangent to a parabola & the perpendicular to it from the focus lies on the tangent at the vertex. (D) P & Q are the points with eccentric angles & + on the ellipse x a triangle OPQ is independent of. y =, then the area of the b []

13 Q.0 For the hyperbola x y = the incorrect statement is : 9 (A) the acute angle between its asymptotes is 60º (B) its eccentricity is 4/ (C) length of the latus rectum is (D) product of the perpendicular distances from any point on the hyperbola on its asymptotes is less than the length of its latus rectum. Q.04 The locus of the mid point of the focal radii of a variable point moving on the parabola, y = 4ax is a parabola whose (A) Latus rectum is half the latus rectum of the original parabola (B) Vertex is (a/, 0) (C) Directrix is y-axis (D) Focus has the co-ordinates (a, 0) Q.05 P is a point on the parabola y = 4ax (a > 0) whose vertex is A. PA is produced to meet the directrix in D and M is the foot of the perpendicular from P on the directrix. If a circle is described on MD as a diameter then it intersects the xaxis at a point whose coordinates are : (A) ( a, 0) (B) (a, 0) (C) ( a, 0) (D) (a, 0) Q.06 If the circle x + y = a intersects the hyperbola xy = c in four points P(x, y ), Q(x, y ), R(x, y ), S(x 4, y 4 ), then (A) x + x + x + x 4 = 0 (B) y + y + y + y 4 = 0 (C) x x x x 4 = c 4 (D) y y y y 4 = c 4 Q.07 Extremities of the latera recta of the ellipses (a > b) having a given major axis a lies on a b (A) x = a(a y) (B) x = a (a + y) (C) y = a(a + x) (D) y = a (a x) Q.08 Let y = 4ax be a parabola and x + y + bx = 0 be a circle. If parabola and circle touch each other externally then : (A) a > 0, b > 0 (B) a > 0, b < 0 (C) a < 0, b > 0 (D) a < 0, b < 0 Q.09 The tangent to the hyperbola, x y = at the point, 0 when associated with two asymptotes constitutes : (A) isosceles triangle (B) an equilateral triangle (C) a triangles whose area is sq. units (D) a right isosceles triangle. Q.0 Let P, Q and R are three co-normal points on the parabola y = 4ax. Then the correct statement(s) is/are (A) 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 []

14 Q. A variable circle is described to pass through the point (, 0) and tangent to the curve y = tan (tan x). The locus of the centre of the circle is a parabola whose : (A) length of the latus rectum is (B) axis of symmetry has the equation x + y = (C) vertex has the co-ordinates (/4, /4) (D) none of these Q. Which of the following equations in parametric form can represent a hyperbola, where 't' is a parameter. (A) x = a b t & y = t t t (B) tx a y b + t = 0 & x a + ty b = 0 (C) x = e t + e t & y = e t e t (D) x 6 = cos t & y + = 4 cos t Q. The equations of the common tangents to the ellipse, x + 4y = 8 & the parabola y = 4x can be (A) x + y + 4 = 0 (B) x y + 4 = 0 (C) x + y 4 = 0 (D) x y + 4 = 0 Q.4 Variable chords of the parabola y = 4ax subtend a right angle at the vertex. Then : (A) 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 Equations of a common tangent to the two hyperbolas x a (A) y = x + a y = & y b a b (B) y = x a b x = is : b (C) y = x + a b (D) x a b Q.6 The equation of the tangent to the parabola y = (x ) parallel to the chord joining the points (, 0) and (4, ) is : (A) x y + 6 = 0 (B) y x + 6 = 0 (C) 4 y 4 x + = 0 (D) 4 x 4 y = Q.7 Let A be the vertex and L the length of the latus rectum of the parabola, y y 4 x 7 = 0. The equation of the parabola with A as vertex, L the length of the latus rectum and the axis at right angles to that of the given curve is : (A) x + 4 x + 8 y 4 = 0 (B) x + 4 x 8 y + = 0 (C) x + 4 x + 8 y + = 0 (D) x + 8 x 4 y + 8 = 0 Q.8 The differential equation dx dy = y x of lines) with eccentricity : represents a family of hyperbolas (except when it represents a pair (A) 5 (B) 5 (C) 5 (D) 5 [4]

15 Q.9 If a number of ellipse be described having the same major axis a but a variable minor axis then the tangents at the ends of their latera recta pass through fixed points which can be (A) (0, a) (B) (0, 0) (C) (0, a) (D) (a, a) Q.0 The straight line y + x = touches the parabola : (A) x + 4 y = 0 (B) x x + y = 0 (C) 4 x x + y = 0 (D) x x + y = 0 Q. Circles are drawn on chords of the rectangular hyperbola xy = c parallel to the line y = x as diameters. All such circles pass through two fixed points whose co-ordinates are : (A) (c, c) (B) (c, c) (C) ( c, c) (D) ( c, c) [5]

16 Select the correct alternative : (Only one is correct) Q. C Q. B Q. B Q.4 D Q.5 C Q.6 A Q.7 B Q.8 B Q.9 A Q.0 B Q. C Q. B Q. C Q.4 B Q.5 B Q.6 B Q.7 C Q.8 D Q.9 B Q.0 D Q. B Q. A Q. D Q.4 A Q.5 D Q.6 C Q.7 B Q.8 B Q.9 B Q.0 A Q. A Q. D Q. D Q.4 B Q.5 B Q.6 C Q.7 A Q.8 A Q.9 B Q.40 B Q.4 D Q.4 C Q.4 C Q.44 B Q.45 C Q.46 B Q.47 D Q.48 B Q.49 C Q.50 A Q.5 D Q.5 C Q.5 C Q.54 C Q.55 D Q.56 B Q.57 A Q.58 B Q.59 C Q.60 C Q.6 C Q.6 B Q.6 A Q.64 C Q.65 A Q.66 C Q.67 B Q.68 D Q.69 B Q.70 B Q.7 C Q.7 A Q.7 D Q.74 A Q.75 C Q.76 B Q.77 C Q.78 A Q.79 D Q.80 B Q.8 D Q.8 C Q.8 D Q.84 C Q.85 A Q.86 B Q.87 D Q.88 A Q.89 C Q.90 B Q.9 D Q.9 C Q.9 D Q.94 C Q.95 B Q.96 A Q.97 D Q.98 D Q.99 C Q.00 D Select the correct alternatives : (More than one are correct) Q.0 A,B Q.0 A,C,D Q.0 B,D Q.04 A,B,C,D Q.05 A,D Q.06 A,B,C,D Q.07 A,B Q.08 A,D Q.09 B,C Q.0 A,B,C,D Q. B,C Q. A,C,D Q. A,B Q.4 A,B,C Q.5 A,B,C,D Q.6 C,D Q.7 A,B Q.8 B,D Q.9 A,C Q.0 A,B,C Q. A,D ANSWER KEY [6]