Find the Vertices 25x^2+9y^2-150x+72y+144=0

$25 x^{2} + 9 y^{2} - 150 x + 72 y + 144 = 0$

Step 1

Find the standard form of the ellipse.

Subtract $144$ from both sides of the equation.

$25 x^{2} + 9 y^{2} - 150 x + 72 y = - 144$

Complete the square for $25 x^{2} - 150 x$ .

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 $25 {(x - 3)}^{2} - 225$ .

$25 {(x - 3)}^{2} - 225$

Substitute $25 {(x - 3)}^{2} - 225$ for $25 x^{2} - 150 x$ in the equation $25 x^{2} + 9 y^{2} - 150 x + 72 y = - 144$ .

$25 {(x - 3)}^{2} - 225 + 9 y^{2} + 72 y = - 144$

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

$25 {(x - 3)}^{2} + 9 y^{2} + 72 y = - 144 + 225$

Complete the square for $9 y^{2} + 72 y$ .

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 $9 {(y + 4)}^{2} - 144$ .

$9 {(y + 4)}^{2} - 144$

Substitute $9 {(y + 4)}^{2} - 144$ for $9 y^{2} + 72 y$ in the equation $25 x^{2} + 9 y^{2} - 150 x + 72 y = - 144$ .

$25 {(x - 3)}^{2} + 9 {(y + 4)}^{2} - 144 = - 144 + 225$

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

$25 {(x - 3)}^{2} + 9 {(y + 4)}^{2} = - 144 + 225 + 144$

Simplify $- 144 + 225 + 144$ .

Add $- 144$ and $225$ .

$25 {(x - 3)}^{2} + 9 {(y + 4)}^{2} = 81 + 144$

Add $81$ and $144$ .

$25 {(x - 3)}^{2} + 9 {(y + 4)}^{2} = 225$

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

$\frac{25 {(x - 3)}^{2}}{225} + \frac{9 {(y + 4)}^{2}}{225} = \frac{225}{225}$

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 $1$ .

$\frac{{(x - 3)}^{2}}{9} + \frac{{(y + 4)}^{2}}{25} = 1$

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.

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

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.

$a = 5$

$b = 3$

$k = - 4$

$h = 3$

Step 4

Find the vertices.

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

$(h, k + a)$

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

$(3, - 4 + 5)$

Simplify.

$(3, 1)$

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

$(h, k - a)$

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

$(3, - 4 - (5))$

Simplify.

$(3, - 9)$

Ellipses have two vertices.

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

${Vertex}_{2}$ : $(3, - 9)$

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

${Vertex}_{2}$ : $(3, - 9)$

Step 5

Find the Vertices 25x^2+9y^2-150x+72y+144=0

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