by Abhijit Kumar Jha

Size: px

Start display at page:

Download "by Abhijit Kumar Jha"

Kathleen Watkins
5 years ago
Views:

1 SET I. If the locus of the point of intersection of perpendicular tangents to the ellipse x a circle with centre at (0, 0), then the radius of the circle would e a + a /a ( a ). There are exactl two points on the ellipse x a are equal and equal to a Ahijit Kumar Jha is a whose distance from the center of the ellipse. Eccentricit of this ellipse is equal to. If the line = mx + c, interests the ellipse x a, at points whose eccentric angles differ /, then (a m + ) = 4c (a + m ) = 4c a m + = 4c a + m = 4c 4. Consider an ellipse with major and minor axes of length 0 and 8 units respectivel. The radius of largest circle that can e inscried in this ellipse, it is given that centre of this circle is one focus of the ellipse, is equal to 4 units units 6 units none of these. Eccentricit of the ellipse x + 6x + = 8 is 6. The tangent at the point '' on the ellipse x + = meets the auxiliar circle in two a points which sutends a right angle at the centre, then the eccentricit 'e' of the ellipse is given the equation e ( + cos ) = e. (cosec ) = e ( + sin ) = e ( + tan ) = 7. S and T are the foci of an ellipse and B is an end of the minor axis. If STB is equilateral, then e is

2 /4 / / none of these Ahijit Kumar Jha 8. A ladder units long slides in a vertical plane with its ends in contact with a vertical wall and a horizontal floor along x-axis. The locus of a point on the ladder 4 units from its foot has the equation : x 4 + = x = 6 64 x = x = 9. Eccentric angle of a point on the ellipse x + = 6 at a distance units from the centre of the ellipse is / / / 4 none of these 0. The point of intersection of the tangents at the point P on the ellipse x a corresponding point Q on the auxiliar circle meet on the line : x = a/e x = 0 = 0 none of these. If and are eccentric angles of the ends of a focal chord of the ellipse x a tan tan e e e e is equal to e e e e = and its, then. The distances from the foci of P(a, ) on the ellipse x 9 are 4 4 a 4 4 none of these. If tan. tan = a x then the chord joining two points & on the ellipse a

3 will sutend a right angle at focus end of the major axis centre end of the minor axis Ahijit Kumar Jha 4. The equations of the common tangents to the ellipse, x + 4 = 8 & the paraola = 4x are x = ± x 4 = ± x + 4 = ± none of these. If O is the centre, OA the semimajor axis and S the focus of an ellipse, the eccentric angle of an point P is POS PSA PAS none of these 6. If A and B are two fixed points and P is a variale point such that PA + PB = 4, the locus of P is a paraola an ellipse a hperola none of these F HG 7. If P( ) and Q I K J are two points one the ellipse x a x x a a x a 4 none of these, locus of mid-point of PQ is 8. The length of the chord of the ellipse x where mid-point is none of these F HG, 9. The sum of the square of perpendiculars on an tangent to the ellipse x from two point a on the minor axis, each at a distance are from the centre, is a a + a - 0. If latus rectum of the ellipse x tan sec is / then ( 0 ) is equal to / / 6 / 8 none of these I K J

4 SET II Ahijit Kumar Jha. The length of the major axis of the ellipse (x 0) + ( + ) = (x 4 7) 4 4 is. The tangent and normal to the ellipse x + 4 = 4 at a point P( ) on it meets the major axes in Q and R respectivel. If QR =, then cos is equal to 4 none of these. x The ellipse and the straight line = mx + c intersect in real points onl if a a m < c a m > c a m c c 4. The foci of the ellipse (x + ) + 9 ( + ) = are at (, ) and (, 6) (, ) and (, 6) (, ) and (, ) (, ) and (, 6).. The parametric representation of a point on the ellipse whose foci are (, 0) and (7, 0) and eccentricit / is ( + 8cos, 4 sin ) (8cos, 4 sin ) ( + 4 cos, 8sin ) none of these 6. x The equation =, will represent an ellipse if 6 a a a (, ) a(, 6) a (, ) (6, ) a (, 6) ~ {4} 7. Tangents are drawn to the ellipse x + = 4 from an aritrar point on the line x + = 4, the corresponding chord of contact will alwas pass through a fixed point, whose coordinates are,, 8. The line = x touches the ellipse x + 4 =, at,,

5 , Ahijit Kumar Jha (, ) (, ) None of these x 9. The normal drawn to the ellipse at the extremit of the latus rectum passes through the a extremit of the minor axis. Eccentricit of this ellipse is equal to x 0. The line x = 8 is a normal to the ellipse. If e the eccentric angle of the foot 9 of this normal, then is equal to None of these 6 4 x. Tangent drawn to the ellipse at point P meets the coordinate axes at points A and B a respectivel. Locus of mid-point of segment AB is x a a x 4 4 a x x a x. Tangents PA and PB are drawn to the ellipse from the point P(0, ). Area of triangle 6 9 PAB is equal to sq. units sq. units sq. units sq. units x. Tangents are drawn to the ellipse from an point on the paraola = 4x. The 6 9 corresponding chord of contact will touch a paraola, whose equation is + 4x = 0 = 4x + 9x = 0 = 9x 4. The normal at a variale point P on an ellipse x = of eccentricit e meets the axes of the a 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 e = e = e e = /e

6 Ahijit Kumar Jha. An ellipse is such that the length of the latus rectum is equal to the sum of the lengths of its semi principal axes. Then ulges to a circle ecomes a line segment etween the two foci ecomes a paraola none of these 6. If the line x + 4 = 7 touches the ellipse x + 4 = then, the point of contact is 7, 7, 7, 7 none of these 7. A common tangent to 9x + 6 = 44 ; x + 4 = 0 & x + x + = 0 is = x = 4 x = 4 = 8. If F & F are the feet of the perpendiculars from the foci S & S of an ellipse x = on the tangent at an point P on the ellipse, then (S F ). (S F ) is equal to 4 9. The area of the rectangle formed the perpendiculars from the centre of the standard ellipse to the tangent and normal at its point whose eccentric angle is /4 is a a a a a a a a a a a a 0. If & are the eccentric angles of the extremities of a focal chord of an standard ellipse, then the eccentricit of the ellipse is : cos cos cos( ) sin sin sin( ) cos cos cos( ) sin sin sin( )

/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2

/ 3, then (A) 3(a 2 m 2 + b 2 ) = 4c 2 (B) 3(a 2 + b 2 m 2 ) = 4c 2 (C) a 2 m 2 + b 2 = 4c 2 (D) a 2 + b 2 m 2 = 4c 2 SET I. If the locus of the point of intersection of perpendiculr tngents to the ellipse x circle with centre t (0, 0), then the rdius of the circle would e + / ( ) is. There re exctl two points on the