Math Problem Solver

\(11 {x}^{2} + 8 {y}^{2} - 4 x + 28 y - 16 = 0\) is an equation of an ellipse, and is equivalent to \(\frac{242 \left(x - \frac{2}{11}\right)^{2}}{899} + \frac{176 \left(y + \frac{7}{4}\right)^{2}}{899} = 1\).

It is centered at \(\left(\frac{2}{11}, - \frac{7}{4}\right)\).

Its foci are \(\left(\frac{2}{11}, - \frac{7}{4} + \frac{\sqrt{2697}}{44}\right)\) and \(\left(\frac{2}{11}, - \frac{7}{4} - \frac{\sqrt{2697}}{44}\right)\).

Its minor axis has length \(\frac{\sqrt{1798}}{11}\) with endpoints \(\left(\frac{2}{11} - \frac{\sqrt{1798}}{22}, - \frac{7}{4}\right)\) and \(\left(\frac{2}{11} + \frac{\sqrt{1798}}{22}, - \frac{7}{4}\right)\).

Its major axis has length \(\frac{\sqrt{9889}}{22}\) with endpoints \(\left(\frac{2}{11}, - \frac{\sqrt{9889}}{44} - \frac{7}{4}\right)\) and \(\left(\frac{2}{11}, - \frac{7}{4} + \frac{\sqrt{9889}}{44}\right)\).

The ellipse has parameterization: \[\begin{aligned}x\left(\theta\right) &= \frac{2}{11} + \frac{\sqrt{1798} \sin{\left(\theta \right)}}{22}\\y\left(\theta\right) &= - \frac{7}{4} + \frac{\sqrt{9889} \cos{\left(\theta \right)}}{44}\end{aligned}\]