Question

write an equation of the ellipse with the given characteristics and center at \\((0, 0)\\).

vertex: \\((0, 4)\\)

focus: \\((0, 2\sqrt{3})\\)

equation:

Explanation:

Step1: Identify ellipse orientation

The vertex \((0,4)\) and focus \((0,2\sqrt{3})\) lie on the y-axis, indicating a vertical major axis.

Step2: Determine semi-major axis

The distance from the center \((0,0)\) to the vertex \((0,4)\) gives \(a = 4\).

Step3: Determine focal distance

The distance from the center \((0,0)\) to the focus \((0,2\sqrt{3})\) gives \(c = 2\sqrt{3}\).

Step4: Calculate semi-minor axis

Using \(b^2 = a^2 - c^2\), we find \(b^2 = 4^2 - (2\sqrt{3})^2 = 16 - 12 = 4\).

Step5: Write the equation

Substitute \(a^2 = 16\) and \(b^2 = 4\) into the standard vertical ellipse equation \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\).

Answer:

$$\frac{x^2}{4} + \frac{y^2}{16} = 1$$