x^2+4y^2+2x+8y+1=0
You asked:
Investigate the equation: \({x}^{2} + 4 \cdot {y}^{2} + 2 x + 8 y + 1 = 0\).
MathBot Answer:
\({x}^{2} + 4 \cdot {y}^{2} + 2 x + 8 y + 1 = 0\) is an equation of an ellipse, and is equivalent to \(\frac{\left(x + 1\right)^{2}}{4} + 1 \left(y + 1\right)^{2} = 1\).
It is centered at \(\left(-1, -1\right)\).
Its foci are \(\left(- \sqrt{3} - 1, -1\right)\) and \(\left(-1 + \sqrt{3}, -1\right)\).
Its minor axis has length \(2\) with endpoints \(\left(-1, -2\right)\) and \(\left(-1, 0\right)\).
Its major axis has length \(4\) with endpoints \(\left(-3, -1\right)\) and \(\left(1, -1\right)\).
The ellipse has parameterization: \[\begin{aligned}x\left(\theta\right) &= -1 + 2 \cos{\left(\theta \right)}\\y\left(\theta\right) &= -1 + \sin{\left(\theta \right)}\end{aligned}\]