Question

Let $OABC$ be a rectangle, where $O$ is the origin and $A$, $C$ lie on the parabola $y=x^2$. If $B$ lies on a parabola whose vertex coordinates is $(\alpha ,\beta )$, then the value of $\alpha +\beta$ is

Answer 1

Let the coordinates of $A$ be $(t,t^2)$
and coordinates of $C$ be $(a,a^2)$.
$OA\perp OC$
$m_{OA}\times m_{OC}=-1\Rightarrow \frac{t^2-0}{t-0}\times \frac{a^2-0}{a-0}=-1\Rightarrow a=-\frac{1}{t}$
$\therefore$ Coordinates of $C$ is $(-\frac{1}{t},\frac{1}{t^2})$

Let coordinates of $B$ be $(h,k)$.
Since, the diagonals $OB$ and $AC$ bisect each other, we have
$(\frac{h}{2},\frac{k}{2})=(\frac{t-\frac{1}{t}}{2},\frac{t^2+\frac{1}{t^2}}{2})$
$\Rightarrow h=t-\frac{1}{t},k=t^2+\frac{1}{t^2}$

Now, $h^2=t^2+\frac{1}{t^2}-2$
$\Rightarrow h^2=k-2$
Locus of point $B$ is $y=x^2+2$
Vertex of the parabola is $(\alpha ,\beta )=(0,2)$
$\Rightarrow \alpha +\beta =2$