Question

A tangent is drawn at any point $(x_1,y_1)$ other than the vertex on the parabola $y^2=4ax$. Tangents are drawn from any point of this tangent to the circle $x^2+y^2=a^2$ such that all the chords of contact passes 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)

$x_1{x}_2+y_1{y}_2=a^2$

Let $(x_1,y_1)=(a{t}^2,2at)$
Tangent at this point is
$ty=x+a{t}^2$
Any point on this tangent will be
$(h,\frac{h+a{t}^2}{t})$ .
Chord of contact drawn from this point on the circle is $T=0$
$\Rightarrow hx+\frac{(h+a{t}^2)}{t}y=a^2$
$\Rightarrow aty-a^2+h(x+\frac{y}{t})=0$
It represents the family of straight lines through the points of intersection of
$ty-a=0$ and $x+\frac{y}{t}=0$
So, the fixed point is $(x_2,y_2)=(-\frac{a}{t^2},\frac{a}{t})$

$(x_1,y_1)=(a{t}^2,2at)$
Clearly, $x_1{x}_2=-a^2$
So, $x_1,a,x_2$ are not in $G.P.$

$\frac{y_1{y}_2}{2}=a^2$
So, $\frac{y_1}{2},a,y_2$ are in $G.P.$

$\frac{x_1}{x_2}=-t^4,\frac{y_1}{y_2}=2t^2$
$-4\times \frac{x_1}{x_2}={\left(\frac{y_1}{y_2}\right)}^2$
So, $-4,\frac{y_1}{y_2},\frac{x_1}{x_2}$ are in $G.P.$

$x_1{x}_2+y_1{y}_2=-a^2+2a^2=a^2$