Questions: Write an equation of the ellipse with the given characteristics and center at (0,0). Vertex: (3,0) Co-Vertex: (0,2) Equation:

Solution Steps

To find the equation of the ellipse, we need to identify the lengths of the semi-major axis and the semi-minor axis. The vertex at $(3,0)$ indicates that the semi-major axis is along the x-axis with length 3, so $a = 3$. The co-vertex at $(0,2)$ indicates that the semi-minor axis is along the y-axis with length 2, so $b = 2$. The standard form of the ellipse centered at the origin is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

The vertex of the ellipse is given as \((3,0)\), which indicates that the length of the semi-major axis \(a\) is 3. The co-vertex is given as \((0,2)\), indicating that the length of the semi-minor axis \(b\) is 2.

The standard form of an ellipse centered at the origin \((0,0)\) is given by the equation: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] Substituting the values of \(a\) and \(b\): \[ \frac{x^2}{3^2} + \frac{y^2}{2^2} = 1 \]

Calculating \(a^2\) and \(b^2\): \[ a^2 = 3^2 = 9 \quad \text{and} \quad b^2 = 2^2 = 4 \] Thus, the equation simplifies to: \[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]

Final Answer