Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse from any of its foci?
Explanation for the correct option:
Step 1: Compare with the given equation.
Step 2: Find the eccentricity.
The formula for the eccentricity of an ellipse is
Step 3: Find the focus.
Focus is given as .
Focus [Rationalizing the denominator]
Step 4: Derive the equation of the tangent.
The equation of the tangent is given as
The line passes through .
Step 5: Derive the equation of the perpendicular.
The slope of the perpendicular is given as .
The equation of the perpendicular line is given as
Step 6: Add both the equations and to find the locus.
[Equation of a circle]
Step 7: Put the coordinates in the equation.
Since the left-hand side is equal to the right-hand side, lies on the locus of the foot of the perpendicular.
The correct answer is option a) .