TITLE: Equation of the locus of the foot of perpendicular from any focus upon any tangent to the ellipse ${x^2\over a^2}+{y^2\over b^2}=1$
QUESTION [7 upvotes]: Find the equation of the locus of the foot of perpendicular from any focus upon any tangent to the ellipse ${x^2\over a^2}+{y^2\over b^2}=1$.
will it also be an ellipse?

REPLY [3 votes]: Foot of perpendicular drawn from focus to any tangent lies on the auxiliary circle of the ellipse hence the locus will be a circle and not an ellipse. Moreover you may also try to verify the property by finding the the foot of perpendicular and putting it into the equation:
$$
x^2 + y^2 = a^2 \qquad\text{(equation of auxillary circle for a standard ellipse)}
$$
where $a$ is the length of semi-major axis.