Question

the ellipse with x-intercepts (5,0) and (-5,0); y-intercepts (0,3) and (0,-3)

Answer 1

Given an ellipse with following parameters

$\begin{gathered} x-\text{intercepts }\Rightarrow(5,0),(-5,0) \\ y-\text{intercepts}\Rightarrow(0,3),(0,-3) \end{gathered}$

The general formula of an ellipse is given as

$\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1$

The coordinate of the center is (h, k )

The vertices on x-axis is at (b, 0) and (-b , 0)

The vertices on y-axis is at (0, a) and (0, -a)

We can deduce from the parameters provided that

$\begin{gathered} h=0,k=0 \\ (b,0)=(5,0)\Rightarrow b=5 \\ (0,a)=(0,3)\Rightarrow a=3 \end{gathered}$

Thus the ellipse equation can be calculated as

$\begin{gathered} \frac{(x-0)^2}{5^2}+\frac{(y-0)^2}{3^2}=1 \\ \frac{x^2}{25}+\frac{y^2}{9}=1 \end{gathered}$

Hence, the ellipse equation is

$\frac{x^2}{25}+\frac{y^2}{9}=1$