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Q.
The locus of the centre of the circle described on any focal chord of the parabola $y^{2}=4ax$ as the diameter is
NTA AbhyasNTA Abhyas 2022
Solution:
Let, $P\left(a t_{1}^{2} , 2 a t_{1}\right)$ and $Q\left(a t_{2}^{2} , 2 a t_{2}\right)$ be the extremities of a focal chord $PQ$ of the parabola $y^{2}=4ax.$ Then, $t_{1}t_{2}=-1.$
Let, $\left(h , k\right)$ be the coordinates of the centre of the circle described on $PQ$ as diameter. Then,
$h=\frac{a}{2}\left(t_{1}^{2} + t_{2}^{2}\right)$ and $k=a\left(t_{1} + t_{2}\right)$
$\Rightarrow \frac{2 h}{a}=t_{1}^{2}+t_{2}^{2}$ and $\left(\frac{k}{a}\right)^{2}=\left(t_{1} + t_{2}\right)^{2}$
$\Rightarrow \frac{2 h}{a}=t_{1}^{2}+t_{2}^{2}$ and $\frac{k^{2}}{a^{2}}=t_{1}^{2}+t_{2}^{2}+2t_{1}t_{2}$
$\Rightarrow \frac{k^{2}}{a^{2}}=\frac{2 h}{a}-2$ $\left[\because t_{1} t_{2} = - 1\right]$
$\Rightarrow k^{2}=2a\left(h - a\right)$
Hence, the locus of $\left(h , k\right)$ is
$y^{2}=2a\left(x - a\right).$