Mathematics

Mathematics

Problem

An ellipse has its major axis along the xx-axis. The distance between its foci is 1010 and the length of its minor axis is 88. (i) Determine the standard equation of the ellipse. (ii) Find the eccentricity of the ellipse. (iii) Write the coordinates of the foci and the vertices. (iv) Obtain the coordinates of the endpoints of the latus rectum. (v) Calculate the exact length of the latus rectum.
Intuitive explanation

An ellipse is defined by the relationship between its semi-major axis aa, semi-minor axis bb, and the distance of each focus from center cc, where c2=a2−b2c^2 = a^2 - b^2. The key is extracting these three values from the given information—the distance between foci tells you 2c2c, and the minor axis length gives you 2b2b—then using them to build the standard form and unlock all the geometric properties like eccentricity, foci positions, and the special chord through each focus perpendicular to the major axis.

Intuitive explanation

An ellipse is defined by the relationship between its semi-major axis aa, semi-minor axis bb, and the distance of each focus from center cc, where c2=a2−b2c^2 = a^2 - b^2. The key is extracting these three values from the given information—the distance between foci tells you 2c2c, and the minor axis length gives you 2b2b—then using them to build the standard form and unlock all the geometric properties like eccentricity, foci positions, and the special chord through each focus perpendicular to the major axis.

Browse the problem library