6x² + 12y² - 24x - 144y - 360 = 0

asked by guest
on Nov 28, 2024 at 12:16 am



You asked:

Investigate the equation: \(6 {x}^{2} + 12 {y}^{2} - 24 x - 144 y - 360 = 0\).

MathBot Answer:

\(6 {x}^{2} + 12 {y}^{2} - 24 x - 144 y - 360 = 0\) is an equation of an ellipse, and is equivalent to \(\frac{\left(x - 2\right)^{2}}{136} + \frac{\left(y - 6\right)^{2}}{68} = 1\).

It is centered at \(\left(2, 6\right)\).

Its foci are \(\left(2 - 2 \sqrt{17}, 6\right)\) and \(\left(2 + 2 \sqrt{17}, 6\right)\).

Its minor axis has length \(4 \sqrt{17}\) with endpoints \(\left(2, 6 - 2 \sqrt{17}\right)\) and \(\left(2, 6 + 2 \sqrt{17}\right)\).

Its major axis has length \(4 \sqrt{34}\) with endpoints \(\left(2 - 2 \sqrt{34}, 6\right)\) and \(\left(2 + 2 \sqrt{34}, 6\right)\).

The ellipse has parameterization: \[\begin{aligned}x\left(\theta\right) &= 2 + 2 \sqrt{34} \cos{\left(\theta \right)}\\y\left(\theta\right) &= 6 + 2 \sqrt{17} \sin{\left(\theta \right)}\end{aligned}\]

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