Find the Vertices 9x^2+25y^2-72x+250y+544=0

$9 x^{2} + 25 y^{2} - 72 x + 250 y + 544 = 0$

Step 1

Find the standard form of the ellipse.

Subtract $544$ from both sides of the equation.

$9 x^{2} + 25 y^{2} - 72 x + 250 y = - 544$

Complete the square for $9 x^{2} - 72 x$ .

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

$a = 9$

$b = - 72$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- 72}{2 \cdot 9}$

Simplify the right side.

Cancel the common factor of $- 72$ and $2$ .

Factor $2$ out of $- 72$ .

$d = \frac{2 \cdot - 36}{2 \cdot 9}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 36}{2 (9)}$

Cancel the common factor.

$d = \frac{2 \cdot - 36}{2 \cdot 9}$

Rewrite the expression.

$d = \frac{- 36}{9}$

Cancel the common factor of $- 36$ and $9$ .

Factor $9$ out of $- 36$ .

$d = \frac{9 \cdot - 4}{9}$

Cancel the common factors.

Factor $9$ out of $9$ .

$d = \frac{9 \cdot - 4}{9 (1)}$

Cancel the common factor.

$d = \frac{9 \cdot - 4}{9 \cdot 1}$

Rewrite the expression.

$d = \frac{- 4}{1}$

Divide $- 4$ by $1$ .

$d = - 4$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 72)}^{2}}{4 \cdot 9}$

Simplify the right side.

Simplify each term.

Raise $- 72$ to the power of $2$ .

$e = 0 - \frac{5184}{4 \cdot 9}$

Multiply $4$ by $9$ .

$e = 0 - \frac{5184}{36}$

Divide $5184$ by $36$ .

$e = 0 - 1 \cdot 144$

Multiply $- 1$ by $144$ .

$e = 0 - 144$

Subtract $144$ from $0$ .

$e = - 144$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(x - 4)}^{2} - 144$ .

$9 {(x - 4)}^{2} - 144$

Substitute $9 {(x - 4)}^{2} - 144$ for $9 x^{2} - 72 x$ in the equation $9 x^{2} + 25 y^{2} - 72 x + 250 y = - 544$ .

$9 {(x - 4)}^{2} - 144 + 25 y^{2} + 250 y = - 544$

Move $- 144$ to the right side of the equation by adding $144$ to both sides.

$9 {(x - 4)}^{2} + 25 y^{2} + 250 y = - 544 + 144$

Complete the square for $25 y^{2} + 250 y$ .

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

$a = 25$

$b = 250$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{250}{2 \cdot 25}$

Simplify the right side.

Cancel the common factor of $250$ and $2$ .

Factor $2$ out of $250$ .

$d = \frac{2 \cdot 125}{2 \cdot 25}$

Cancel the common factors.

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

$d = \frac{2 \cdot 125}{2 (25)}$

Cancel the common factor.

$d = \frac{2 \cdot 125}{2 \cdot 25}$

Rewrite the expression.

$d = \frac{125}{25}$

Cancel the common factor of $125$ and $25$ .

Factor $25$ out of $125$ .

$d = \frac{25 \cdot 5}{25}$

Cancel the common factors.

Factor $25$ out of $25$ .

$d = \frac{25 \cdot 5}{25 (1)}$

Cancel the common factor.

$d = \frac{25 \cdot 5}{25 \cdot 1}$

Rewrite the expression.

$d = \frac{5}{1}$

Divide $5$ by $1$ .

$d = 5$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{250^{2}}{4 \cdot 25}$

Simplify the right side.

Simplify each term.

Raise $250$ to the power of $2$ .

$e = 0 - \frac{62500}{4 \cdot 25}$

Multiply $4$ by $25$ .

$e = 0 - \frac{62500}{100}$

Divide $62500$ by $100$ .

$e = 0 - 1 \cdot 625$

Multiply $- 1$ by $625$ .

$e = 0 - 625$

Subtract $625$ from $0$ .

$e = - 625$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $25 {(y + 5)}^{2} - 625$ .

$25 {(y + 5)}^{2} - 625$

Substitute $25 {(y + 5)}^{2} - 625$ for $25 y^{2} + 250 y$ in the equation $9 x^{2} + 25 y^{2} - 72 x + 250 y = - 544$ .

$9 {(x - 4)}^{2} + 25 {(y + 5)}^{2} - 625 = - 544 + 144$

Move $- 625$ to the right side of the equation by adding $625$ to both sides.

$9 {(x - 4)}^{2} + 25 {(y + 5)}^{2} = - 544 + 144 + 625$

Simplify $- 544 + 144 + 625$ .

Add $- 544$ and $144$ .

$9 {(x - 4)}^{2} + 25 {(y + 5)}^{2} = - 400 + 625$

Add $- 400$ and $625$ .

$9 {(x - 4)}^{2} + 25 {(y + 5)}^{2} = 225$

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

$\frac{9 {(x - 4)}^{2}}{225} + \frac{25 {(y + 5)}^{2}}{225} = \frac{225}{225}$

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 $1$ .

$\frac{{(x - 4)}^{2}}{25} + \frac{{(y + 5)}^{2}}{9} = 1$

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.

$a = 5$

$b = 3$

$k = - 5$

$h = 4$

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.

$(4 + 5, - 5)$

Simplify.

$(9, - 5)$

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.

$(4 - (5), - 5)$

Simplify.

$(- 1, - 5)$

Ellipses have two vertices.

${Vertex}_{1}$ : $(9, - 5)$

${Vertex}_{2}$ : $(- 1, - 5)$

${Vertex}_{1}$ : $(9, - 5)$

${Vertex}_{2}$ : $(- 1, - 5)$

Step 5

Find the Vertices 9x^2+25y^2-72x+250y+544=0

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