Question

If points $(a{u}^2,2au)$ and $(a{v}^2,2av)$ are extremities of the focal chord of a parabola $y^2=4ax,$ then

• A. $u+v=0$
• B. $u-v=0$
• C. $uv-1=0$
• D. $uv+1=0$

Answer 1

The correct option is D $uv+1=0$

Coordinates of extremities of focal chord are $(x_1,y_1)=(a{u}^2,2au)$ and $(x_2,y_2)=(a{v}^2,2av)$ and equation of parabola is $y^2=4ax.$

We know that coordinates of the focus $(x,y)=(a,0).$

We know that, the equation of the focal chord with $(x_1,y_1)$ and $(x_2,y_2)$ as extremity points is $(y-y_1)=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$

$\Rightarrow (y-2au)=\frac{2av-2au}{{av}^2-{au}^2}(x-{au}^2)\Rightarrow y-2au=\frac{2}{v+u}(x-{au}^2)$

Since the focal chord passes through the focus, therefore $0-2au=\frac{2}{v+u}(a-{au}^2)$

$\Rightarrow -uv-u^2=1-u^2$

$\Rightarrow uv+1=0.$