Question

If $(x_r,y_r):r=1,2,3,4$ be the points of intersection of the parabola $y^2=4ax$ and the circle $x^2+y^2+2gx+2fy+c=0,$ then

• A. $y_1+y_2+y_3+y_4=0$
• B. $\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}=0$
• C. $y_1-y_2+y_3-y_4=0$
• D. $y_1+y_2-y_3-y_4=0$

Answer 1

Answer: (A), (B)

$\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}=0$

Intersection points are $(x_r,y_r):r=1,2,3,4$
$y^2=4ax$ and the circle is $x^2+y^2+2gx+2fy+c=0$
From parabola equation, we have
$x=\frac{y^2}{4a}$

Now, $\frac{y^4}{16a^2}+y^2+\frac{2g{y}^2}{4a}+2fy+c=0$
Above equation has four roots.
As coefficient of $y^3$ is $0,$
so, $y_1+y_2+y_3+y_4=0$

$y=\pm \sqrt{4ax}$
Clearly, $\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}=0$