Objective Mathematics

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1 . A tangent to the ellipse is intersected by a b the tangents at the etremities of the major ais at 'P' and 'Q' circle on PQ as diameter always passes through : (a) one fied point two fied points (c) four fied points (d) three fied points. A variable tangent of ellipse meets the a b co-ordinate aes at A and B minimum area (in sq. units) of circumcircle of AOB, 'O' being the origin, is given by : (a) ( ) 4 a b. ( a b ). (c) ( ) 4 a b. (d) ( ) 4 a b.. Let P (, y) be any point on ellipse 9 + 5y = 5, if 'F ' and 'F ' are the focal points of ellipse perimeter of F PF is : (a) 0 8 (c) 5 (d) 0 4. The chords of contact of tangents t o curve + 8y = 8 from any point on its director circle intersect the director circle at 'C' and 'D' locus of the point of intersection of tangents to circle at 'C' and 'D' is : (a) 6 + y = y = 4. (c) 64 + y = 6. (d) None of these. 5. If normal at an end of latus rectum of an ellipse passes through one etremity of minor ais eccentricity 'e' satisfy : (a) e 4 + e = 0 e + e 5 = 0 (c) e = 5/ (d) e 4 e + = 0 6. If tangent is drawn at ' ' point to the ellipse 7 y 7, where 0, value of ' ' such that sum of intercepts on aes made by this tangent is minimum, is : (a) 8 (c) 6 (d) 4 7. The length of latus rectum of an ellipse is one third of the major ais eccentricity of ellipse is equal to : (a) (c) (d) 8. Minimum distance between the ellipse y 6 and the line y 7 0 is equal to : (a) 4 (c) 5 (d) 0 9. The line passing through the etremity A of the major ais and etremity B of the minor ais of the ellipse + 9y = 9 meets its auiliary circle at the point M. Then the area of the triangle with vertices at A, M and the origin O is equal to : (a) 0 (c) 0 sq. units 9 0 sq. unit (d) 7 0 sq. unit sq. units 0. The normal at a point P on the ellipse + 4y = 6 meets the -ais at Q. If M is the mid point of the line segment PQ the locus of M intersects the latus rectums of the given ellipse at the points 5 (a), 7 (c), 7 5 9, 4 4 (d), 7 [ 68 ] Mathematics for JEE-0

2 Ellipse. Maimum length of chord of ellipse, a b a > b, such that eccentric angles of the etremities of chord differ by is : 7. Let normals be drawn to the ellipse + y = from point (, ) the co-normal points lie on the curve : (a) y + 4y = 0 y + 4y = 0 (c) + 4y y = 0 (d) 4y + 4 y = 0 (a) a b (c) ab (d) b a. If an ellipse with major and minor aes of length 0 and 0 units respectively slides along the co-ordinate aes in the first quadrant length of the arc which is formed by the locus of centre of ellipse is given by : (a) (c) 5 (d). Area of ellipse for which focal points are (, 0) and (, 0) and point (4, ) lying on it, is given by : (a) 8 sq. units 9 sq. units (c) 4 sq. units (d) 8 sq. units 4. Let tangents drawn from point 'P' to the ellipse 4y 6 meets the co-ordinate aes at concylic points locus of point 'P' is given by : (a) y = 7 + y = 7 (c) y = 6 (d) + y = 6 5. Let the common tangent in I st quadrant to the circle y 6 and 4 5y 00 meet the aes at A and B area of (a) 4 AOB, where O is origin, is : 8 8. Let 'A' be the centre of ellipse 5 + 5y + 6y 8 = 0 and 'P', 'Q' points lie on the ellipse such that AP and AQ distances are maimum and minimum respectively AP + AQ is equal to : (a) 4 (c) (d) 5 9. Let 'AB' be the variable chord of the ellipse + y = and AOB, where 'O' is origin, then OA OB ( OA. OB) is equal to : (a).. (c) 4. (d) Let normal to the ellipse 4 + 5y = 0 at point P( ) touches the parabola y = 4 tan is equal to : (a) (c) (d) 4. Let circle 'C' with centre (, 0) be inscribed in the ellipse maimum 4y 6 and the area of circle 'C' is (c) 0 (d) none of these 6. Let ellipse, where a > b, be centered at a b 'O' and having AB and CD as its major and minor ais respectively. If one of the focus of ellipse is 'F ', the in-radius of triangle DOF is unit and OF = 6 units director circle of ellipse is given by : (a) + y = 00 + y = 97/ (c) + y = 50 (d) + y = 05/ [ 69 ] (a) equation of director circle of 'C' is given by 9( ) 9y equation of director circle of 'C' is given by ( ) y (c) area of circle 'C' is sq. units. (d) circle 'C' is auiliary circle for the ellipse 9( ) 5y Mathematics for JEE-0

3 . Let one of the focus point of ellipse a y b be at F (4, 0) and its intersection point with positive y-ais be 'B'. If the centre of ellipse is 'C' and circum-radius of CF B is.5 units which of the following statements are incorrect : (a) equation of director circle of ellipse is + y = 4. area of ellipse is 0 square units. (c) director circle of auiliary circle of the ellipse is + y = 50. (d) length of latus rectum of ellipse is 4 units.. Let ellipse E : + 4y = 4 is inscribed in a rectangle aligned with co-ordinate aes, which in turn is inscribed in another ellipse E that passes through the point (4, 0). With reference to ellipse E and E which of the following statements are true : (a) If point (, ) lies in between the boundary of the director circle of E and E 5 5. If point (, ) lies outside the ellipse E R [, ]. (c) Total number of integral points inside the ellipse E are four. (d) If point (, ) lies inside the ellipse E,. 4. Let point 'P' lies on the ellipse and normal 5 6 to ellipse at 'P' meets the co-ordinate aes at A and B. If 'O' is the origin and M is the foot of perpendicular from origin to AB (a) maimum area of AOB is.05 square units. maimum value of OM is units. (c) maimum value of OM is unit. (d) maimum area of AOB is 8 square. units Let variable point 'P' lies on the curve y and PA, PB are tangents to the ellipse + y = 9. If APB is an acute angle co-ordinate of point 'P' can be given by : (a) e e (c) ln (d) 9 tan Following questions are assertion and reasoning type questions. Each of these questions contains two statements, Statement (Assertion) and Statement (Reason). Each of these questions has four alternative answers, only one of them is the correct answer. Select the correct answer from the given options : (a) Both Statement and Statement are true and Statement is the correct eplanation of Statement. Both Statement and Statement are true but Statement is not the correct eplanation of Statement. (c) Statement is true but Statement is false. (d) Statement is false but Statement is true. 6. Statement : Total number of distinct normals which can be drawn to the ellipse point (0, 6) are three. from 69 5 Statement : Maimum number of normals which can be drawn to any given ellipse from a point are four. 7. Let any point 'P' lies on the ellipse and 6 PM, PM are the distances of 'P' from 8 0 and 8 0 respectively. Statement : For point 'P' maimum value of (PM ) (PM ) cannot eceed 64 square units Statement : Area of PF F, where F and F are foci of ellipse, can't eceed 4 square units. 8. Let C and C be two ellipse which are given by + 4y = 4 and + y = 6 respectively and any tangent to curve C meets the curve C at A and B. Statement : If tangents drawn to curve C at points A and B meet at point P APB Statement : Locus of point 'P' is the director circle of curve 'C '. [ 70 ] Mathematics for JEE-0

4 Ellipse 9. Statement : Let 'L' be variable line which is tangential to fied ellipse with foci F and F locus of the foot of perpendicular from foci to line 'L' is the auiliary circle of ellipse Statement : Product of the length of perpendiculars from foci F and F to the line 'L' is always the square of semi-minor ais of ellipse. 0. Statement : If point 'P' lies on a given ellipse with foci at F and F perimeter of PF F is constant Statement : Perimeter of the ellipse is given by ( F F )( e ) e eccentricity of ellipse. units, where 'e' is the [ 7 ] Mathematics for JEE-0

5 Comprehension passage () ( Questions No. - ) Let tangent at any point on the curve E : 4 + 9y = 6 meets the curve E :0 5 y 50 at P and Q. If tangents drawn at P and Q to curve E meets at point 'R' and locus of 'R' is given by the curve 'C ' then answer the following questions.. Locus of point from which perpendicular tangents can be drawn to curve 'C ' is : (a) + y = 50 + y = 60 (c) y 8 = 0 (d) y 9 = 0. Positive slope of the common tangent to curve 'C ' and + y = 60 is : (a) (c) (d). If from any point 'A' on the line + y = 0 tangents AB and AC are drawn to curve 'C ' locus of the circumcentre of ABC is : (a) 4 + 6y = 7 + y = 5 (c) y = 0 (d) + y = 0 Comprehension passage () ( Questions No. 4-6 ) 6. If 'O' is origin and the area of OPQ is square units value of 'k' is (a) (c) 5 4 (d) Comprehension passage () ( Questions No. 7-9 ) Let L : y m = 0 and L : y m = 0 be the variable lines for which m m is negative, and lines L and L are tangential to the variable ellipse 'E' at the points T and T respectively. If the ellipse 'E' is rotating about the point (, 0) and initially its equat ion is given by b ( ) a y ( ab), where R answer the following questions. 7. If 0 and the angle TOT is constant for all the positions of variable ellipse 'E', where 'O' is origin, then the ordered pair (a, b) can be given by : (a) (7, ) (4, 6) (c) (8, 6) (d) (, 6) 8. If a = 4b = and the angle TOT remains acute for all the positions of the variable ellipse 'E', where 'O' is origin the possible value of ' ' can be : (a) e (c) e (d) tan Let variable ellipse + 4y = 4k, where k R, and a fied parabola y = 8 is having a common tangent which meets the co-ordinate aes at P and Q answer the following questions. 4. Let A be the point of contact of the common tangent 9. If the T OT remains obtuse for all the positions of the variable ellipse 'E', where O is origin which one of the following relation must hold true : (a) a b 0 min{ a, b} a b with the ellipse and the eccentric angle of A is, then value of 'k' is equal to : (a) 4 8 (c) 6 (d) 5 (c) (d) ma{ a, b} a b a b a b 5. Locus of the mid-point of the intercepted length PQ is : (a) y + 4 = 0 y + = 0 (c) y + = 0 (d) 4y + = 0 [ 7 ] 0. Let tangent and normal be drawn at any point 'P' on the ellipse + y =, and rectangle PAOB is completed, where 'O' is the origin. Maimum area (in square units) of the rectangle PAOB is... Mathematics for JEE-0

6 . Let common tangents of the curves y = 4 and + 4y = 8 meets on the -ais at A and intersects the positive and negative y-ais at B and C respectively. If parabola with its ais along the -ais and verte at A passes through B and C length of latus rectum of the parabola is... Ellipse 4. Let tangents drawn at A and B points on the ellipse 4 + 9y = 6 meet at point P(, ). If 'C' is the centre of ellipse and the area of quadrilateral PACB is square units value of [ ], where [.] represents the greatest integer function, is equal to.... Let points A, B and C lie on the curve y, 4 y and y respectively maimum value of (AB + AC) is equal to Let ABCD is a square of side length 8 units, and an ellipse of eccentricity 0.5 is drawn touching the sides of the square, where the aes of symmetry being along the diagonals of square. If the major ais and minor ais is of length ' a' and ' b' units respectively. If line y meet the ellipse 4 9y 6 at points 'A' and 'B', where AOB 90, 'O' being the origin positive value of is equal to... value of sec sin b a is Match the following columns (I) and (II) Column (I) Column (II) (a) Number of points on the ellipse + 5y = 00 from (p) 0 which pair of perpendicular tangents can be drawn to the ellipse is / are : (q) 6 9 If the lines y = m + c and y = m + c intersect the ellipse a y at four concyclic points (m b + m ) must be : (r) (c) If all the normals of ellipse intersects or touches 5 6 the circle + y = r minimum value of 'r' is : (s) 4 (d) If the equation + 4y 8 + 6y + 4 k = 0 represents an ellipse values of 'k' can be : [ 7 ] Mathematics for JEE-0

7 7. Let C : + y = a and C : + y = b be two circles, where b > a > 0, and 'O' is origin. A line OPQ is drawn which meets C and C at points P and Q respectively. If 'R' is the moving point for which PR and QR is parallel to the y-ais and -ais respectively and the locus of 'R' is an ellipse 'E' match the following columns for eccentricity 'e' of the ellipse 'E' and the position of foci F and F of 'E'. Column (I) Column (II) (a) If F and F lie on the circle 'C ' eccentricity 'e' can be : If F and F lie inside the circle 'C ' eccentricity 'e' can be : (p) (q) sec sin (c) If F and F lie inside the circle 'C ' eccentricity 'e' can be : (r) cos 4 (d) If F and F don't lie inside the circle 'C ' eccentricity 'e' (s) cos() can be : [ 74 ] Mathematics for JEE-0

8 Ellipse.. (d). 4. (d) 5. (a) E 6. (c) (d) 0. (c). (a). (c). 4. (a) (a) 8. (c) (a). (b, c). (b, d). (a, b) 4. (c, d) 5. (a, b, d) (c) (a). (a). 4. (c) 5. E (c) 8. (c) 9. (c) 0. ( ). ( ). ( 6 ). ( 6 ) 4. ( 7 ) 5. ( 4 ) 6. (a) s 7. (a) r p q, s (c) q (c) p, q, r, s (d) q, r, s (d) p, r [ 75 ] Mathematics for JEE-0

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