The locus of the mid-point of a chord of the circle $x^2+y^2=4$, which subtends a right angle at the origin is
- $x+y=2$
- $x^2+y^2=1$
- $x^2+y^2=2$
- $x+y=1$
Solution
$O C=\sqrt{h^2+k^2}$
In $\triangle O C B$,
$\begin{aligned}
& \cos 45^{\circ}=\frac{\sqrt{h^2+k^2}}{2} \\
& \Rightarrow \quad \frac{\sqrt{h^2+k^2}}{2}=\frac{1}{\sqrt{2}} \\
& \Rightarrow \quad h^2+k^2=2
\end{aligned}$
Replacing $h \rightarrow x$ and $k \rightarrow y$
$\text { Locus } \Rightarrow x^2+y^2=2$Asked in: BITSAT 2024 (Memory Based Paper 3)