PROBLEMS 04 - PARABOLA Page 1

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1 PROBLEMS 0 - PARABOLA Page 1 ( 1 ) Find the co-ordines of the focus, length of the lus-rectum and equion of the directrix of the parabola x - 8. [ Ans: ( 0, - ), 8, ] ( ) If the line x k 0 is a tangent to the parabola 1x, then find k and obtain the co-ordines of the point of contact. 16 Ans : k 16,, - 8 ( ) Derive the equions of the tangents drawn from the point ( 1, ) to the parabola 8x. Obtain the co-ordines of the point of contact. Ans : x (, ) and x 1 1, ( ) Find the equion of the chord of the parabola joining the points P ( t 1 ) and Q ( t ). If this chord passes through the focus, then prove th t 1 t - 1. [ Ans: ( t 1 t ) ( x a t 1 t ) ] ( 5 ) If one end-point of a focal chord of the parabola 16x is ( 9, 1 ), then find its other end-point. 16 Ans :, ( 6 ) The points P ( t 1 ), Q ( t ) and R ( t ) are on the parabola ax. Show th the area of triangle PQR is a l ( t 1 - t ) ( t - t ) ( t - t 1 ) l. ( 7 ) If the focus of the parabola ax divides a focal chord in the rio 1 :, then find the equion of the line containing this focal chord. [ Ans: ± ( x - a ) ]

2 PROBLEMS 0 - PARABOLA Page ( 8 ) If a focal chord of the parabola ax forms an angle of measure θ with the positive X-axis, then show th its length is l a l cosec θ. ( 9 ) Show th the length of the focal chord of the parabola ax the point P ( t ) 1 is l a l t t ( 10 ) Find the condition for the line x cos α sin α p to be a tangent to the parabola ax and obtain the co-ordines of the point of contact. [ Ans: p a sin α tan α 0, ( a tan α, - a tan α ) ] ( 11 ) Show th the equion of the common tangent to the parabolas ax and 1 1 x b is a x b ( ab ) 0. ( 1 ) Find the equions of tangents to the parabola 1x from the point (, 5 ) and the co-ordines of the point of contact. Ans : x - 0, and x - 0 (, 6 ) ( 1 ) The line PA joining a point P on the parabola and the vertex of the parabola intersects the directrix in K. If M is the foot of the perpendicular to the directrix from P, then show th MSK is a right angle. ( 1 ) If the tangent point P of the parabola ax intersects the line x a in K and the directrix in U, then prove th SK SU. ( 15 ) PQ is a focal chord of the parabola ax. The lengths of the perpendicular line segments from the vertex and the focus to the tangents P and Q are p 1, p, p and p respectivel. Show th p 1 p p p a.

3 PROBLEMS 0 - PARABOLA Page ( 16 ) Prove th the orthocentre of the triangle formed b an three tangents to a parabola lies on the directrix. ( 17 ) A tangent of a parabola has a line segment between the tangents the points P and Q. Show th the mid-point of this line segment lies on the tangent parallel to PQ. ( 18 ) If a chord of the parabola ax subtends a right angle the vertex, then show th the point of intersection of the tangents drawn the end-points of this chord is on the line x a 0. ( 19 ) Find the equion of a tangent to the parabola 8x which cuts off equal intercepts along the two axes, and find the co-ordines of the point of contact. [ Ans: x 0, (, - ) ] ( 0 ) Prove th the segment cut out on a tangent to a parabola b the point of contact and the directrix subtends a right angle the focus. ( 1 ) Prove th the foot of the perpendicular from the focus on an tangent to a parabola lies on the Y-axis. ( ) Show th the circle described on an focal chord of a parabola as a diameter touches the directrix. ( ) Prove th, if P is an point on the parabola ax whose focus is S, the circle described on SP as diameter touches the Y-axis. ( ) A quadrileral ABCD is inscribed inside a parabola. If the sides AB, BC, CD and DA of the quadrileral make angles θ 1, θ, θ and θ respectivel with the axis of the parabola, then prove th cot θ 1 cot θ cot θ cot θ.

4 PROBLEMS 0 - PARABOLA Page ( 5 ) Find the points on the parabola 16x which are a distance of 1 units from the focus. [ Ans: ( 9, - 1 ), ( 9, 1 ) ] ( 6 ) Prove th the parabola x divides the line-segment joining ( 1, 1 ) and (, ) internall and externall in the same rio numericall. ( 7 ) Find the measure of the angle between the two tangents drawn from ( 1, ) to the parabola 1x. Ans : 1 tan -1 ( 8 ) Prove th the measure of the angle between the two parabolas x 7 and 8x 9 is tan ( 9 ) If the tangents the points P and Q on the parabola meet T, then prove th ST SP PQ. ( 0 ) Find the point on the parabola 6x which is nearest to the line x 6 0. [ Ans: ( 9, - ) ] ( 1 ) The tangents the points P and Q to the parabola make complementar angles with the axis of the parabola. Prove th the line PQ passes through the point of intersection of the directrix and the axis of the parabola. ( ) The tangents the points P and Q to the parabola with vertex A meet the point T. If the lines AP, AT and AQ intersect the directrix the points P, T and Q respectivel, then prove th PT TQ.

5 PROBLEMS 0 - PARABOLA Page 5 ( ) Prove th the area of the triangle inscribed in the parabola ax is 1 l ( 1 - )( - )( - 1 ) l, where 1, and are the Y-coordines of 8 l a l the vertices. ( ) Prove th the area of the triangle formed b the tangents the parametric points P ( t 1 ), Q ( t ) and R ( t ) to the parabola ax is a l ( t1 - t )( t - t )( t - t1 ) l. ( 5 ) Find the equion of the common tangents to the parabolas x and x. [ Ans: x 0 ] ( 6 ) If ( h, k ) is the point of intersection of the parabolas ax and x b other than the origin, then prove th the equion of their common tangent is ( kx h ) hk 0. ( 7 ) Find the equion of the common tangent to the circle x a and the parabola 8ax. [ Ans: x ± a 0 ] ( 8 ) Find the equion of the line containing the chord of the parabola ax whose midpoint is ( x 1, 1 ). [ Ans: 1-1 a ( x - x1 ) ] ( 9 ) The tangent an point P on the parabola ax meets the X-axis T and the Y-axis R. A is the vertex of the parabola. If RATQ is a rectangle, prove th the locus of the point Q is ax 0. ( 0 ) If the angle between two tangents from point P to the parabola ax is α, then prove th the locus of point P is - ax ( x a ) tan α.

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