Question

If the normals to the curve $y=x^2$ at the points $P, Q$ and $R$ pass through the point $\left(0, \frac{3}{2}\right)$, and the equation of the circle circumscribing the triangle $P Q R$ is $x^2+y^2+2 g x+2 f y+c=0$, then find the value of $\left( g ^2+ f ^2+ c ^2\right)$

Answer 1

$ y = x ^2 ; x = t ; y = t ^2$
$\frac{ dy }{ dx }=2 x =2 t$
$\therefore $ slope of normal $m =\frac{-1}{2 t }$
equation of normal
$y-t^2=-\frac{1}{2 t}(x-t) $ or $ 2 t\left(y-t^2\right)=-x+t$
if $x =0 ; y =\frac{3}{2}$
$2 t \left(\frac{3}{2}- t ^2\right)= t \Rightarrow t =0$
or $3-2 t^2=1 \Rightarrow t=1$ or -1
hence one of the point is origin and the other two are $(-1,1)$ and $(1,1)$

$\Rightarrow PQR$ is a right triangle
$\therefore $ radius of the circle is 1
its equation is $\left.x^2+(y-1)^2=1 \Rightarrow x^2+y^2-2 y=0\right]$