Ellipse Questions with Solutions

An ellipse is the set of points whose distances from two fixed points (the foci) add up to a constant. Its standard form is x^2/a^2 + y^2/b^2 = 1, where a > b, and its eccentricity e satisfies b^2 = a^2(1 - e^2), with 0 < e < 1. The foci are at (ae, 0) and (-ae, 0) and the directrices are x = a/e and x = -a/e. Exam questions test finding the foci, eccentricity, length of the major and minor axes, latus rectum (2b^2/a), focal distances, and equations of tangents and normals. A useful fact is that the sum of the focal distances of any point on the ellipse equals the length of the major axis, 2a. Always check whether the major axis lies along the x-axis or the y-axis before applying formulas. Practise the ellipse questions below, each with a step-by-step solution.