Tangent is drawn at any point `(x_1, y_1)` other than the vertex on the parabola `y^2=4a x` . If tangents are drawn from any point on this tangent to the circle `x^2+y^2=a^2` such that all the chords of contact pass through a fixed point `(x_2,y_2),` then (a) `x_1, a ,x_2` in GP (b) `(y_1)/2,a ,y_2` are in GP
(c)`-4,(y_1)/(y_2),x_1/x_2` are in GP (d) `x_1x_2+y_1y_2=a^2`

Options

A. `x_(1),a,x_(2)` are in GP
B. `(y_(1))/(2),a,y_(2)` are in GP
C. `-4(y_(1))/(y_(2)),(x_(1))/(x_(2))` are in GP
D. `x_(1)x_(2)+y_(1)y_(2)=a^(2)`

Answer

B, C, D

Explanation

2,3,4
Let `(x_(1),y_(1))-=(at^(2),2at)`.
Tangent at this point is `ty=x+at^(2)`.
Any point on this tangent is `((h,(h+at^(2))//t)`.
The chord of contact of this point with respect to the circle
`x^(2)+y^(2)=a^(2)` is
`hx+(h+at^(2))/(t)y=a^(2)`
`or(aty-a^(2))+h(x+(y)/(t))=0`
which is a family of straight lines passing through the point of intersection of
`ty-a=0andx+(y)/(t)=0`
So, the fixed point is `(-a//t^(2),a//t)`. Therefore,
`x_(2)=-(a)/(t^(2)),y_(2)=(a)/(t)`
Clearly, `x_(1)x_(2)=-a^(2),y_(1)y_(2)=2a^(2)`
Also, `(x_(1))/(x_(2))=-t^(4)`
and `(y_(1))/(y_(2))=2t^(2)`
`or4(x_(1))/(x_(2))+((y_(1))/(y_(2)))^(2)=0`