Final answer:
The locus of the foot of the perpendicular drawn from the center to any tangent to the ellipse is a line passing through the center of the ellipse and perpendicular to the tangent. The equation of this line is y = -x/m, where m is the slope of the tangent.
Step-by-step explanation:
The locus of the foot of the perpendicular drawn from the center to any tangent to the ellipse is a line passing through the center of the ellipse and perpendicular to the tangent.
The center of the ellipse is at the origin (0,0) and the equation of the ellipse is x²/a² + y²/b² = 1. Let the equation of the tangent at a point (x₁, y₁) on the ellipse be mx - y + c = 0.
The slope of the tangent is m, and the slope of the line passing through the center is -1/m. Using the point-slope form of a line equation, the equation of the line passing through the center is y - 0 = -1/m(x - 0). Simplifying this equation, we get y = -x/m. By substituting the equation of the tangent into this line equation, we can find the foot of the perpendicular.