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?

• A. $(-1,\sqrt{3})$
• B. $(-2,\sqrt{3})$
• C. $(-1,\sqrt{2})$
• D. $(1,2)$

Answer 1

The correct option is A $(-1,\sqrt{3})$

We know that the locus of foot of the perpendiculars drawn from foci to any tangent of the ellipse is the auxiliary circle.
Auxiliary circle of an ellipse is $x^2+y^2=a^2$
where $a=$ Length of semi-major axis of the ellipse.

Thus the equation of the auxiliary circle of the ellipse $\frac{x^2}{4}+\frac{y^2}{2}=1$ is $x^2+y^2=2^2=4$
From the given options, $(-1,\sqrt{3})$ lies on the auxiliary circle.