Chapter 10.3, Problem 53AYU

In Problems 43-54, analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility.

$4x^2+y^2+4y=0$

To determine

To find: The vertices and foci of the given equation of the ellipse.

Answer to Problem 53AYU

Solution:

Vertices: $\left(0,-4\right)\text{and}\left(0,0\right)$.

Foci: $\left(0,-2-\surd 3\right)\text{and}\left(0,-2+\surd 3\right)$.

Explanation of Solution

Given:

$4x²+y²+4y=0$

Formula used:

Equation	Center	Major axis	Foci	Vertices
$\begin{array}{l}\frac{{\left(x-h\right)}^2}{b^2}+\frac{{\left(y-k\right)}^2}{a^2}=1\\ a>b>0\text{and}b²=a²-c²\end{array}$	$\left(h,k\right)$	$\text{parallel}\text{to}y\text{-axis}$	$\left(h,k+c\right)$	$\left(h,k+a\right)$
	$\left(h,k\right)$	$\text{parallel}\text{to}y\text{-axis}$	$\left(h,k-c\right)$	$\left(h,k-a\right)$

Calculation:

Rearranging the given equation $4x²+y²+4y=0$.

We get, $4\left(x^2\right)+\left(y^2+4y+4\right)=4$.

$4{\left(x\right)}^2+\left(y+2\right)²=4$ dividing by 4, we get,

$x^2+\frac{{\left(y+2\right)}^2}{4}=1$

The equation is of the form $\frac{{\left(x-h\right)}^2}{b^2}+\frac{{\left(y-k\right)}^2}{a²}=1$.

Major axis is parallel to $y\text{-axis}$.

The center of the ellipse is at $(h,k)=\left(0,-2\right);h=0,k=-2$.

$a=2;b=1$

The vertices are $\left(h,k\pm a\right)=\left(0,-2\pm 2\right)$.

Hence, vertices: $\left(0,-4\right)\text{and}\left(0,0\right)$.

To find the value of $c$,

$c²=2²-1²=3;c=\surd 3$

Foci are $\left(h,k\pm c\right)=\left(0,-2\pm \surd 3\right)$.

Hence the foci are $\left(0,-2-\surd 3\right)\text{and}\left(0,-2+\surd 3\right)$.

c.

To graph: The equation $4x²+y²+4y=0$.

Solution:

Given:

$4x²+y²+4y=0$

Calculation:

The graph of the equation of $4x²+y²+4y=0$, is plotted using graphing tool.

Using the given information $4x²+y²+4y=0$, we find the points of the equation as shown in the table below.

Using these points given above, we plot the points and draw a graph. We find center $\left(0,-2\right)$, foci $\left(0,-2-\surd 3\right)\text{and}\left(0,-2+\surd 3\right)$ and vertices $\left(0,-4\right)\text{and}\left(0,0\right)$. We see that the major axis is parallel to $y\text{-axis}$. We plot the graph for the equation using the graphing tool to verify it.