Circle Question 3
Question 3 - 2024 (01 Feb Shift 2)
Let the locus of the mid points of the chords of circle $x^{2}+(y-1)^{2}=1$ drawn from the origin intersect the line $x+y=1$ at $P$ and $Q$. Then, the length of $P Q$ is :
(1) $\frac{1}{\sqrt{2}}$
(2) $\sqrt{2}$
(3) $\frac{1}{2}$
(4) 1
Show Answer
Answer (1)
Solution
$m _{OM} \cdot m _{CM}=-1$
$\frac{k}{h} \cdot \frac{k-1}{h}=-1$
$\therefore$ locus is $x^{2}+y(y-1)=0$
$x^{2}+y^{2}-y=0$ $p=\left|\frac{1 / 2}{\sqrt{2}}\right| p=\frac{1}{2 \sqrt{2}}$
$PQ=2 \sqrt{r^{2}-p^{2}}$
$=2 \sqrt{\frac{1}{4}-\frac{1}{8}}=\frac{1}{\sqrt{2}}$
