Circle 1 Question 25
25. Let $P$ be a point on the circle $S$ with both coordinates being positive. Let the tangent to $S$ at $P$ intersect the coordinate axes at the points $M$ and $N$. Then, the mid-point of the line segment $M N$ must lie on the curve
(a) $(x+y)^{2}=3 x y$
(b) $x^{2 / 3}+y^{2 / 3}=2^{4 / 3}$
(c) $x^{2}+y^{2}=2 x y$
(d) $x^{2}+y^{2}=x^{2} y^{2}$
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Solution:
- We have,
$$ x^{2}+y^{2}=4 $$
Let $P(2 \cos \theta, 2 \sin \theta)$ be a point on a circle.
$\therefore$ Tangent at $P$ is
$2 \cos \theta x+2 \sin \theta y=4$
$\Rightarrow \quad x \cos \theta+y \sin \theta=2$
$\therefore$ The coordinates at $M \frac{2}{\cos \theta}, 0$ and $N 0, \frac{2}{\sin \theta}$
Let $(h, k)$ is mid-point of $M N$ $\therefore \quad h=\frac{1}{\cos \theta} \quad$ and $k=\frac{1}{\sin \theta}$
$\Rightarrow \cos \theta=\frac{1}{h}$ and $\sin \theta=\frac{1}{k}$
$\Rightarrow \quad \cos ^{2} \theta+\sin ^{2} \theta=\frac{1}{h^{2}}+\frac{1}{k^{2}} \Rightarrow 1=\frac{h^{2}+k^{2}}{h^{2} \cdot k^{2}}$
$\Rightarrow h^{2}+k^{2}=h^{2} k^{2}$
$\therefore \quad$ Mid-point of $M N$ lie on the curve $x^{2}+y^{2}=x^{2} y^{2}$
