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Question

# Write the equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hyperbola. Then graph the equation.$9 y^2+18 y=25 x^2+216$

Solution

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Step 1
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Rewrite the given into standard form.

\begin{aligned} 9y^2+18y=25x^2+216 \end{aligned}

Arrange and use APE

\begin{aligned} \left[9y^2+18y\right]-25x^2=216 \end{aligned}

Factor out $9$

\begin{aligned} 9\left[y^2+2y\right]-25x^2=216 \end{aligned}

Complete the square

\begin{aligned} 9\left[y^2+2y+\left(\frac{b}{2}\right)^2\right]-25x^2=216+9\left(\frac{b}{2}\right)^2 \end{aligned}

Substitute

\begin{aligned} 9\left[y^2+2y+\left(\frac{2}{2}\right)^2\right]-25x^2=216+9\left(\frac{2}{2}\right)^2 \end{aligned}

Simplify

\begin{aligned} 9\left[y^2+2y+(1)^2\right]-25x^2&=216+9(1)^2\\ 9(y^2+2y+1)-25x^2&=225 \end{aligned}

Factor

\begin{aligned} 9(y+1)^2-25x^2=225 \end{aligned}

Equate to $1$

\begin{aligned} \frac{9(y+1)^2}{225}-\frac{25x^2}{225}&=\frac{225}{225}\\ \frac{(y+1)^2}{25}-\frac{x^2}{9}&=1 \end{aligned}

Hence, the standard form is $\dfrac{(y+1)^2}{25}-\dfrac{x^2}{9}=1$ and indicates that the graph is Hyperbola.

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