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Let $Q\left(x_1, y_1\right)$ be a variable point and $R(1,0)$ be a point on the circle $x^2+y^2=1$ and $P$ be the mid-point of $Q R$. Then, the locus of the point $P$ is
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Verified Answer
The correct answer is:
$x^2+y^2-x=0$
Let point $P(h, k)$.
$P$ is mid-point of $Q R$.
$$
\therefore h=\frac{x_1+1}{2}, k=\frac{y_1}{2} \Rightarrow x_1=2 h-1, y_1=2 k \text {. }
$$
$\left(x_1, y_1\right)$ lie, on the circle $x^2+y^2=1$
$$
\therefore(2 h-1)^2+(2 k)^2=1 \Rightarrow h^2+k^2-h=0
$$
$\therefore$ Locus of the point $P$ is $x^2+y^2-x=0$
$P$ is mid-point of $Q R$.
$$
\therefore h=\frac{x_1+1}{2}, k=\frac{y_1}{2} \Rightarrow x_1=2 h-1, y_1=2 k \text {. }
$$
$\left(x_1, y_1\right)$ lie, on the circle $x^2+y^2=1$
$$
\therefore(2 h-1)^2+(2 k)^2=1 \Rightarrow h^2+k^2-h=0
$$
$\therefore$ Locus of the point $P$ is $x^2+y^2-x=0$
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