Question

A tangent is drawn at any point $(x_1,y_1)$ other than vertex on the parabola $y^2=4ax$. 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. $x_1,a,x_2$ are in G.P
• B. $\frac{y_1}{2},a,y_2$ are in G.P.
• C. $-4,\frac{y_1}{y_2},\frac{x_1}{x_2}$ are in G.P.
• D. $x_1{x}_2+y_1{y}_2=a^2$

Answer 1

Answer: (B), (C), (D)

The correct options are
B $\frac{y_1}{2},a,y_2$ are in G.P.
C $-4,\frac{y_1}{y_2},\frac{x_1}{x_2}$ are in G.P.
D $x_1{x}_2+y_1{y}_2=a^2$

Let $(x_1,y_1)\equiv (a{t}^2,2at)$
Tangent at this point $\Rightarrow ty=x+a{t}^2$
Any point on this tangent $\equiv (h,\left(\frac{h+a{t}^2}{t}\right))$ Chord of contact w.r.t the circle $x^2+y^2=a^2$
$hx+\frac{(h+a{t}^2)}{t}y=a^2$
$\Rightarrow (aty-a^2)+h(x+\frac{y}{t})=0$
$ty-a=0$ or $x+\frac{y}{t}=0$
$\therefore$ fixed point $\equiv (\frac{-a}{t^2},\frac{a}{t})$
$\therefore x_1{x}_2=-a^2,Y_1{Y}_2=2a^2$
$\frac{x_1}{x_2}=-t^{4}and\frac{y_1}{y_2}=2t^2$
$\Rightarrow 4\frac{x_1}{x_2}+{\left(\frac{y_1}{y_2}\right)}^2=0$