Question

If the tangent to ellipse $x^{2}+2 y=1$ at point $P\left(\frac{1}{\sqrt{2}}, \frac{1}{2}\right)$ meets the auxiliary circle at the points $R$ and $Q$, then tangents to circle at $Q$ and $R$ intersect at

• A. $ \left( \frac{1}{\sqrt{2}},\,1 \right) $
• B. $ \left( 1,\,\frac{1}{\sqrt{2}} \right) $
• C. $ \left( \frac{1}{2},\,\frac{1}{2} \right) $
• D. $ \left( \frac{1}{2},\,\frac{1}{\sqrt{2}} \right) $

Answer 1

Answer: (A)

Equation of tangent to ellipse at given point is
$x\left(\frac{1}{\sqrt{2}}\right)+2 y\left(\frac{1}{2}\right)=1$
$\Rightarrow x+\sqrt{2} y=\sqrt{2}$ ... (i)

Now, $Q R$ is chord of contact of circle $x^{2}+y^{2}=1$
with respect to point $T(h, K)$.
then, $Q R \equiv h x +K y=1$ ... (ii)
Equations (I) and (li) represent same straight line, then comparing ratio of coefficients we have
$\frac{h}{1}=\frac{K}{\sqrt{2}}=\frac{1}{\sqrt{2}}$
Hence; $(h, K) \equiv\left(\frac{1}{\sqrt{2}}, 1\right)$