Which of the following points lies on the locus
Question:

Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse,

$\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$ from any of its foci?

1. (1) $(-2, \sqrt{3})$

2. (2) $(-1, \sqrt{2})$

3. (3) $(-1, \sqrt{3})$

4. (4) $(1,2)$

Correct Option: 3,

Solution:

We know that the locus of the feet of the perpendicular

draw from foci to any tangent of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ is

the auxiliary circle $x^{2}+y^{2}=a^{2}$

$\therefore$ Auxiliary circle $: x^{2}+y^{2}=4$

$\therefore$ Auxiliary circle $: x^{2}+y^{2}=4$

$\therefore(-1, \sqrt{3})$ satisfies the given equation.