Question

A variable tangent to the parabola $y^2=4ax$ meets the circle $x^2+y^2=r^2$ at P & Q .Then the locus of the mid point of PQ is $x\left(x^2+y^2\right)+a{y}^2$.

This statement is ___.

• A. True
• B. False

Answer 1

The correct option is A True

Equation of tangent to parabola $y^2=4ax$ at point $(x_1,y_1)$ is given by $y{y}_1=2a(x+x_1)\to$ ---(1)

Also $(x_1,y_1)$ lies on parabola

$\Rightarrow y_1^2=4a{x}_1\to$ (2)

Equation of given circle is

$x^2+y^2=r^2.$

Let (h,k) be the mid point of PQ

$\because$ equation of chord whose mid point is (h,k) is given by $T=x_1$ i.e $xh+yk=h^2+k^2\to$ (3)

$equation$ equation (1) and (3) are identical.

Therefore $\frac{2a}{h}=-\frac{y_1}{k}=\frac{2ax}{-(h^2+k^2)}$

$\Rightarrow y_1=\frac{-2ak}{h}\to$ (4)

and $x_1=-\frac{(h^2+k^2)}{h}\to$ (5)

putting the value of $x_1$ and $y_1$ from equation (4) and (5) is (2).

we get, ${\left(\frac{-2ak}{h}\right)}^2=4a\left(\frac{-(h^2+k^2)}{h}\right)$

$\Rightarrow \frac{4a^2{k}^2}{h^2}=-\frac{4a}{h}(h^2+k^2)$

$4ah(h^2+k^2)+4a^2{k}^2=0$

$\Rightarrow ah(h^2+k^2)+a^2{k}^2=0$

$\Rightarrow$ Locus of mid-point of PQ is given by $x(x^2+y^2)+a{y}^2=0$

$\therefore$ This statement is True.