Question

The locus of mid point of that chord of parabola which subtends right angle on the vertex will be

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

Equation of parabola $y^2=4ax.....\left(i\right)$
Equation of that chord of parabola whose mid point is $x_1,y_1$ will be $y{y}_1-2d(x+x_1)=y_1^2-4a{x}_1$
or $y{y}_1-2ax=y_1^2-2a{x}_1$ or $\frac{}{}=1$ .....(ii)

Making equation (i)homogeneous by equation (ii), the equation of lines joining the vertex(0,0) of parabola to the point of intersection of chord (ii) and parabola (i) will be

$y^2=4ax\frac{y{y}_1-2ax}{y_1-2a{x}_1}or{y}^2(y_1^2-2a{x}_1)=4ax(y{y}_1-2ax)or8a^2+(y_1^2-2a{x}_1)=0$ or $y_1^2-2a{x}_1+8a^2=0$
If lines represented by it are mutually perpendicular, then coefficient of $x^2$ + coefficient of $y^2=0$
$\therefore$ Required locus of $(x_1,y_1)$ is $y^2-2ax+8a^2=0$