Chapter 12.1, Problem 60E

To determine

To calculate: The coordinates of center, foci and vertices of the ellipse,

  $4x^2+25y^2+250y+525=0$ . Also sketch the graph.

Answer to Problem 60E

The coordinates of centerare $\left(0,-5\right)$ , foci are $\left(\pm \sqrt{21},-5\right)$ and vertices are $\left(\pm 5,-5\right)$ , $\left(0,-3\right)\text{ and }\left(0,-7\right)$ . The graph is provided below,

Explanation of Solution

Given information:

The equation of ellipse $4x^2+25y^2+250y+525=0$ .

Formula used:

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

The coordinates of center are $\left(h,k\right)$ .

Relation: $c^2=a^2-b^2$ .

The coordinates of foci are $\left(h\pm c,k\right)$ .

The major axis is $y=k$ and minor axis $x=h$ .

The major axis vertices are $\left(h\pm a,k\right)$ .

The minor axis vertices are $\left(h,k\pm b\right)$ .

Calculation:

Consider the equation of ellipse $4x^2+25y^2+250y+525=0$ .

Rewrite it in the standard form. Use the complete the square method.

  $\begin{array}{c}4x^2+25y^2+250y+525=0\\ 4x^2+25\left(y^2+10y\right)+525=0\\ 4x^2+25{\left(y+5\right)}^2-625+525=0\\ 4x^2+25{\left(y+5\right)}^2=100\end{array}$

Simplify it further as,

$\begin{array}{c}\frac{4}{100}{x}^2+\frac{25{\left(y+5\right)}^2}{100}=1\\ \frac{{\left(x-0\right)}^2}{25}+\frac{{\left(y+5\right)}^2}{4}=1\\ \frac{{\left(x-0\right)}^2}{5^2}+\frac{{\left(y-\left(-5\right)\right)}^2}{2^2}=1\end{array}$

Recall that the standard form of ellipse is $\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1$ .

The coordinates of center are $\left(h,k\right)$ .

Relation: $c^2=a^2-b^2$ .

The coordinates of foci are $\left(h\pm c,k\right)$ .

The major axis is $y=k$ and minor axis $x=h$ .

The major axis vertices are $\left(h\pm a,k\right)$ .

The minor axis vertices are $\left(h,k\pm b\right)$ .

Apply it, here $h=0,k=-5,a=5\text{}\text{ and }b=2$ .

Therefore,

  $\begin{array}{c}{c}^2=25-4\\ c^2=21\\ c=\pm \sqrt{21}\end{array}$

The coordinates of center are $\left(0,-5\right)$ .

The coordinates of foci are $\left(\pm \sqrt{21},-5\right)$ .

The major axis vertices are $\left(\pm 5,-5\right)$ .

The minor axis vertices are $\left(0,-3\right)\text{ and }\left(0,-7\right)$ .

Sketch the graph of the ellipse $4x^2+25y^2+250y+525=0$ .

Thus, the coordinates of center are $\left(0,-5\right)$ , foci are $\left(\pm \sqrt{21},-5\right)$ and vertices are $\left(\pm 5,-5\right)$ , $\left(0,-3\right)\text{ and }\left(0,-7\right)$ .