Chapter 10, Problem 3RE

In Problems 1-10, identify each equation. If it is a parabola, give its vertex, focus, and directrix; if it is an ellipse, give its center, vertices, and foci: if it is a hyperbola, give its center, vertices, foci, and asymptotes.

$\frac{y^2}{25}+\frac{x^2}{16}\text{ = }1$

To determine

Each equation. If it is a parabola, give its vertex, focus, and directrix; if it is an ellipse, give its center, vertices, and foci; if it is a hyperbola, give its center, vertices, foci, and asymptotes.

Answer to Problem 3RE

Ellipse

Center: $\left(0,0\right)$

Vertices: $\left(0,\pm 5\right)$

Foci: $\left(0,\pm 3\right)$.

Explanation of Solution

Given:

$\frac{y^2}{25}+\frac{x^2}{16}=1$

Formula used:

Equation	Center	Major axis	Foci	Vertices
$\frac{y^2}{a^2}+\frac{x^2}{b^2}=1$ $a>b>0$ and $b²=a²-c²$	$\left(0,0\right)$	$y\text{-axis}$	$\left(0,\mathrm{c}\right)$	$\left(0,\mathrm{a}\right)$
	$\left(0,0\right)$	$y\text{-axis}$	$\left(0,-c\right)$	$\left(0,-a\right)$

Calculation:

The equation $\frac{y^2}{25}+\frac{x^2}{16}=1$ is of the form $\frac{x^2}{b^2}+\frac{y^2}{a²}=1$ which represents an ellipse.

$a=5,b=\text{ 4}$

Center at origin $\left(0,0\right)$.

Vertices: $\left(0,\pm a\right)=\left(0,\pm 5\right)$.

$c^2=a^2-b^2$;

$c=\sqrt{25-16}=3$

Foci: $\left(0,\pm c\right)=\left(0,\pm 3\right)$.