Find the Vertices 9x^2+25y^2-18x+150y+9=0

$9 x^{2} + 25 y^{2} - 18 x + 150 y + 9 = 0$

Step 1

Find the standard form of the ellipse.

Subtract $9$ from both sides of the equation.

$9 x^{2} + 25 y^{2} - 18 x + 150 y = - 9$

Complete the square for $9 x^{2} - 18 x$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 9$

$b = - 18$

$c = 0$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 18}{2 \cdot 9}$

Simplify the right side.

Cancel the common factor of $- 18$ and $2$ .

Factor $2$ out of $- 18$ .

$d = \frac{2 \cdot - 9}{2 \cdot 9}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 9$ .

$d = \frac{2 \cdot - 9}{2 (9)}$

Cancel the common factor.

$d = \frac{2 \cdot - 9}{2 \cdot 9}$

Rewrite the expression.

$d = \frac{- 9}{9}$

Cancel the common factor of $- 9$ and $9$ .

Factor $9$ out of $- 9$ .

$d = \frac{9 \cdot - 1}{9}$

Cancel the common factors.

Factor $9$ out of $9$ .

$d = \frac{9 \cdot - 1}{9 (1)}$

Cancel the common factor.

$d = \frac{9 \cdot - 1}{9 \cdot 1}$

Rewrite the expression.

$d = \frac{- 1}{1}$

Divide $- 1$ by $1$ .

$d = - 1$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 18)}^{2}}{4 \cdot 9}$

Simplify the right side.

Simplify each term.

Raise $- 18$ to the power of $2$ .

$e = 0 - \frac{324}{4 \cdot 9}$

Multiply $4$ by $9$ .

$e = 0 - \frac{324}{36}$

Divide $324$ by $36$ .

$e = 0 - 1 \cdot 9$

Multiply $- 1$ by $9$ .

$e = 0 - 9$

Subtract $9$ from $0$ .

$e = - 9$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(x - 1)}^{2} - 9$ .

$9 {(x - 1)}^{2} - 9$

Substitute $9 {(x - 1)}^{2} - 9$ for $9 x^{2} - 18 x$ in the equation $9 x^{2} + 25 y^{2} - 18 x + 150 y = - 9$ .

$9 {(x - 1)}^{2} - 9 + 25 y^{2} + 150 y = - 9$

Move $- 9$ to the right side of the equation by adding $9$ to both sides.

$9 {(x - 1)}^{2} + 25 y^{2} + 150 y = - 9 + 9$

Complete the square for $25 y^{2} + 150 y$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 25$

$b = 150$

$c = 0$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{150}{2 \cdot 25}$

Simplify the right side.

Cancel the common factor of $150$ and $2$ .

Factor $2$ out of $150$ .

$d = \frac{2 \cdot 75}{2 \cdot 25}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 25$ .

$d = \frac{2 \cdot 75}{2 (25)}$

Cancel the common factor.

$d = \frac{2 \cdot 75}{2 \cdot 25}$

Rewrite the expression.

$d = \frac{75}{25}$

Cancel the common factor of $75$ and $25$ .

Factor $25$ out of $75$ .

$d = \frac{25 \cdot 3}{25}$

Cancel the common factors.

Factor $25$ out of $25$ .

$d = \frac{25 \cdot 3}{25 (1)}$

Cancel the common factor.

$d = \frac{25 \cdot 3}{25 \cdot 1}$

Rewrite the expression.

$d = \frac{3}{1}$

Divide $3$ by $1$ .

$d = 3$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{150^{2}}{4 \cdot 25}$

Simplify the right side.

Simplify each term.

Raise $150$ to the power of $2$ .

$e = 0 - \frac{22500}{4 \cdot 25}$

Multiply $4$ by $25$ .

$e = 0 - \frac{22500}{100}$

Divide $22500$ by $100$ .

$e = 0 - 1 \cdot 225$

Multiply $- 1$ by $225$ .

$e = 0 - 225$

Subtract $225$ from $0$ .

$e = - 225$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $25 {(y + 3)}^{2} - 225$ .

$25 {(y + 3)}^{2} - 225$

Substitute $25 {(y + 3)}^{2} - 225$ for $25 y^{2} + 150 y$ in the equation $9 x^{2} + 25 y^{2} - 18 x + 150 y = - 9$ .

$9 {(x - 1)}^{2} + 25 {(y + 3)}^{2} - 225 = - 9 + 9$

Move $- 225$ to the right side of the equation by adding $225$ to both sides.

$9 {(x - 1)}^{2} + 25 {(y + 3)}^{2} = - 9 + 9 + 225$

Simplify $- 9 + 9 + 225$ .

Add $- 9$ and $9$ .

$9 {(x - 1)}^{2} + 25 {(y + 3)}^{2} = 0 + 225$

Add $0$ and $225$ .

$9 {(x - 1)}^{2} + 25 {(y + 3)}^{2} = 225$

Divide each term by $225$ to make the right side equal to one.

$\frac{9 {(x - 1)}^{2}}{225} + \frac{25 {(y + 3)}^{2}}{225} = \frac{225}{225}$

Simplify each term in the equation in order to set the right side equal to $1$ . The standard form of an ellipse or hyperbola requires the right side of the equation be $1$ .

$\frac{{(x - 1)}^{2}}{25} + \frac{{(y + 3)}^{2}}{9} = 1$

Step 2

This is the form of an ellipse. Use this form to determine the values used to find the center along with the major and minor axis of the ellipse.

$\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - k)}^{2}}{b^{2}} = 1$

Step 3

Match the values in this ellipse to those of the standard form. The variable $a$ represents the radius of the major axis of the ellipse, $b$ represents the radius of the minor axis of the ellipse, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from the origin.

$a = 5$

$b = 3$

$k = - 3$

$h = 1$

Step 4

Find the vertices.

The first vertex of an ellipse can be found by adding $a$ to $h$ .

$(h + a, k)$

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(1 + 5, - 3)$

Simplify.

$(6, - 3)$

The second vertex of an ellipse can be found by subtracting $a$ from $h$ .

$(h - a, k)$

Substitute the known values of $h$ , $a$ , and $k$ into the formula.

$(1 - (5), - 3)$

Simplify.

$(- 4, - 3)$

Ellipses have two vertices.

${Vertex}_{1}$ : $(6, - 3)$

${Vertex}_{2}$ : $(- 4, - 3)$

${Vertex}_{1}$ : $(6, - 3)$

${Vertex}_{2}$ : $(- 4, - 3)$

Step 5

Find the Vertices 9x^2+25y^2-18x+150y+9=0

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