Question

graph the ellipse.
25x² + 9y² - 150x + 36y + 36 = 0

Explanation:

Step1: Complete the square for x and y terms.

Group the x - terms and y - terms:
\[

$$\begin{align*} 25x^{2}-150x + 9y^{2}+36y&=- 36\\ 25(x^{2}-6x)+9(y^{2}+4y)&=-36 \end{align*}$$

\]
Complete the square for \(x^{2}-6x=(x - 3)^{2}-9\) and \(y^{2}+4y=(y + 2)^{2}-4\).
\[

$$\begin{align*} 25((x - 3)^{2}-9)+9((y + 2)^{2}-4)&=-36\\ 25(x - 3)^{2}-225+9(y + 2)^{2}-36&=-36\\ 25(x - 3)^{2}+9(y + 2)^{2}&=225 \end{align*}$$

\]

Step2: Write the equation in standard form.

Divide both sides of the equation \(25(x - 3)^{2}+9(y + 2)^{2}=225\) by 225:
\(\frac{(x - 3)^{2}}{9}+\frac{(y + 2)^{2}}{25}=1\)
The center of the ellipse is \((h,k)=(3,-2)\), \(a = 5\) (semi - major axis along the y - axis since \(a^{2}=25\)), \(b = 3\) (semi - minor axis along the x - axis since \(b^{2}=9\)).

Step3: Find the key points for graphing.

Vertices:
The vertices along the major axis (parallel to the y - axis) are \((3,-2 + 5)=(3,3)\) and \((3,-2-5)=(3,-7)\).
The vertices along the minor axis (parallel to the x - axis) are \((3 + 3,-2)=(6,-2)\) and \((3-3,-2)=(0,-2)\).

Answer:

Graph an ellipse with center \((3,-2)\), vertices \((3,3)\), \((3,-7)\), \((6,-2)\), \((0,-2)\) on the given coordinate - plane.