11x²+8y²-4x+28y+16=0

asked by guest
on Nov 18, 2024 at 2:22 am



You asked:

Investigate the equation: \(11 \cdot {x}^{2} + 8 \cdot {y}^{2} - 4 x + 28 y + 16 = 0\).

MathBot Answer:

\(11 \cdot {x}^{2} + 8 \cdot {y}^{2} - 4 x + 28 y + 16 = 0\) is an equation of an ellipse, and is equivalent to \(\frac{242 \left(x - \frac{2}{11}\right)^{2}}{195} + \frac{176 \left(y + \frac{7}{4}\right)^{2}}{195} = 1\).

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

Its foci are \(\left(\frac{2}{11}, - \frac{7}{4} + \frac{3 \sqrt{65}}{44}\right)\) and \(\left(\frac{2}{11}, - \frac{7}{4} - \frac{3 \sqrt{65}}{44}\right)\).

Its minor axis has length \(\frac{\sqrt{390}}{11}\) with endpoints \(\left(\frac{2}{11} - \frac{\sqrt{390}}{22}, - \frac{7}{4}\right)\) and \(\left(\frac{2}{11} + \frac{\sqrt{390}}{22}, - \frac{7}{4}\right)\).

Its major axis has length \(\frac{\sqrt{2145}}{22}\) with endpoints \(\left(\frac{2}{11}, - \frac{7}{4} - \frac{\sqrt{2145}}{44}\right)\) and \(\left(\frac{2}{11}, - \frac{7}{4} + \frac{\sqrt{2145}}{44}\right)\).

The ellipse has parameterization: \[\begin{aligned}x\left(\theta\right) &= \frac{2}{11} + \frac{\sqrt{390} \sin{\left(\theta \right)}}{22}\\y\left(\theta\right) &= - \frac{7}{4} + \frac{\sqrt{2145} \cos{\left(\theta \right)}}{44}\end{aligned}\]

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