List of top Mathematics Questions on Ellipse Geometry

Let a focus of the ellipse E: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ be S(4, 0) and its eccentricity be $\frac{4}{5}$. If the point P(3, $\alpha$) lies on E and O is the origin, then the area of $\Delta$POS is equal to:

Let an ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\quad a<b \] pass through the point \((4,3)\) and have eccentricity \( \frac{\sqrt5}{3} \). Then the length of its latus rectum is:

The sum of squares of all the real solutions of the equation \(\log_{(x+1)}(2x^2 + 5x + 3) = 4 - \log_{(2x+3)}(x^2 + 2x + 1)\) is equal to ________.

The length of the chord of the ellipse: \[ \frac{x^2}{4} + \frac{y^2}{2} = 1, \] whose mid-point is \( \left( 1, \frac{1}{2} \right) \), is: