Question

The equation of the ellipse having vertices at $(\pm 5,0)$ and foci $(\pm 4,0)$ is:

• A. $\frac{x^2}{25}+\frac{y^2}{16}=1$
• B. $9x^2+25y^2=225$
• C. $\frac{x^2}{9}+\frac{y^2}{25}=1$
• D. $4x^2+5y^2=20$

Answer 1

The correct option is B

$9x^2+25y^2=225$

Explanation for the correct answer:

Finding the equation of the ellipse,

Given that Vertices are $(5,0)\text{and}(-5,0)$

Foci are $(4,0)\text{and}(-4,0)$

We know that the vertices of an ellipse are given by $(\pm a,0)$

Therefore, $a=5$

Also, the foci of an ellipse is given by: $(\pm c,0)$

Therefore, $c=4$

Calculating $b$:

$$\begin{gathered} c^2=a^2-b^2 \\ \Rightarrow b^2=a^2-c^2 \\ \Rightarrow b^2=25-16 \\ \Rightarrow b=3 \end{gathered}$$

Calculating the equation of the ellipse

$$\begin{gathered} \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \\ \Rightarrow \frac{x^2}{25}+\frac{y^2}{9}=1 \\ \Rightarrow 9x^2+25y^2=225 \end{gathered}$$

Hence, the correct answer is Option (B).