Chapter 10.4, Problem 71AYU

In Problems 67-74, analyze each equation.

$25x^2+9y^2-250x+400=0$

To determine

To find: Analyze the equation.

Answer to Problem 71AYU

The equation represents Ellipse.

Explanation of Solution

Given:

$25x^2+9y^2-250x+400=0$

Formula used:

Equation	Vertex	Focus	Major axis	Description
$\frac{{\left(y-k\right)}^2}{a^2}+\frac{{\left(x-h\right)}^2}{b^2}=1$ $a>b>0 \text{and} b^2=a^2-c^2$	$\left(h,k\pm a\right)$	$\left(h,k\pm c\right)$	$y\text{-axis}$	Major axis is parallel to $y\text{-axis}$.

Calculation:

Given that $25x^2+9y^2-250x+400=0$.

$25\left(x^2-10x\right)+9y²+400=0$

$25{\left(x-5\right)}^2+9y²+400=625$

$25{\left(x-5\right)}^2+9y²=225$

$\frac{25{\left(x-5\right)}^2}{225}+\frac{9y^2}{225}=1$

$\frac{y^2}{25}+\frac{{\left(x-5\right)}^2}{9}=1$

The equation represents an ellipse.

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

Center: $\left(h,k\right)=\left(5,0\right)$.

Vertices: $V\left(h,k\pm a\right)=\left(5,\pm 5\right)$.

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

$c²=5²-3²$; $c=4$

Foci: $F\left(h,k\pm c\right)=F\left(5,\pm 4\right)$.

The graph of the equation of $25x²+9y²-250x+400=0$ is plotted.

Using the given equation $25x²+9y²-250x+400=0$, we plot the graph using graphing tool.

The graph represents an ellipse, with vertex $V\left(5,5\right)$ and $V\left(5,-4\right)$, the Major axis is parallel to $y\text{-axis}$.