Question

Consider a tangent to the ellipse \(\frac{x^{2}}{2}+\frac{y^{2}}{1}=1\) at any point. The locus of the midpoint of the portion intercepted between the axes is
(A) \(\frac{x^{2}}{2}+\frac{y^{2}}{4}=1\)
(B) \(\frac{x^{2}}{4}+\frac{y^{2}}{2}=1\)
(C) \(\frac{1}{3 x^{2}}+\frac{1}{4 y^{2}}=1\)
(D) \(\frac{1}{2 x^{2}}+\frac{1}{4 y^{2}}=1\)

Answer 1

Ans: (D)
Hint: Tangent at \(P\left(x_{1}, y_{1}\right)\)
$$
\frac{x x_{1}}{2}+\frac{y y_{1}}{1}=1
$$
Let mid point of intercept be \(\mathrm{P}(\mathrm{h}, \mathrm{k})\)
$$
\mathrm{h}=\frac{1}{\mathrm{x}_{1}}, \mathrm{k}=\frac{1}{2 \mathrm{y}_{1}} \text { or } \mathrm{x}_{1}=\frac{1}{\mathrm{~h}}, \mathrm{y}_{1}=\frac{1}{2 \mathrm{k}}
$$
locus is \(\frac{1}{2 x^{2}}+\frac{1}{4 y^{2}}=1\)