Question

A tangent to the parabola $y^2=4ax$ meets the axes at A and B. Then the locus of mid point of $AB$ is

• A. $y^2+2ax=0$
• B. $y^2-2ax=0$
• C. $y^2+ax=0$
• D. $2y^2+ax=0$

Answer 1

The correct option is D $2y^2+ax=0$

R.E.F.image.

A] at $(x_1,y_1)$

Tangent, $y{y}_1=2a(x+x_1)$

$y\left(2ax\right)=2a(x+a{x}^2)$

or $xy=x+a{x}^2$

x $y=\frac{x}{t}+at$

= 0 at

$=-a{t}^2$ 0

$\therefore A=(0,at)$ $B(-a{t}^2,0)$

C is mid point of AB

$\therefore (\frac{0-a{t}^2}{2},\frac{0+at}{2})=(\frac{-a{t}^2}{2},\frac{at}{2})=(h,k)$

at D

$y^2=4ax$

$=(2at{)}^2=4a(a{t}^2)$

$=(2\times 2k{)}^2=4a(-2h)$

$=16k^2=-8ah$

$=2k^2+ah=0$

$\therefore 2y^2+ax=0$ [locus of midpoint of AB]