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