Find the Vertices 25x^2+16y^2-50x+160y+25=0

$25 x^{2} + 16 y^{2} - 50 x + 160 y + 25 = 0$

Step 1

Find the standard form of the ellipse.

Subtract $25$ from both sides of the equation.

$25 x^{2} + 16 y^{2} - 50 x + 160 y = - 25$

Complete the square for $25 x^{2} - 50 x$ .

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

$a = 25$

$b = - 50$

$c = 0$

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

$d = \frac{- 50}{2 \cdot 25}$

Simplify the right side.

Cancel the common factor of $- 50$ and $2$ .

Factor $2$ out of $- 50$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor $25$ out of $- 25$ .

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

Cancel the common factors.

Factor $25$ out of $25$ .

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

Cancel the common factor.

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

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

$e = 0 - \frac{{(- 50)}^{2}}{4 \cdot 25}$

Simplify the right side.

Simplify each term.

Raise $- 50$ to the power of $2$ .

$e = 0 - \frac{2500}{4 \cdot 25}$

Multiply $4$ by $25$ .

$e = 0 - \frac{2500}{100}$

Divide $2500$ by $100$ .

$e = 0 - 1 \cdot 25$

Multiply $- 1$ by $25$ .

$e = 0 - 25$

Subtract $25$ from $0$ .

$e = - 25$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $25 {(x - 1)}^{2} - 25$ .

$25 {(x - 1)}^{2} - 25$

Substitute $25 {(x - 1)}^{2} - 25$ for $25 x^{2} - 50 x$ in the equation $25 x^{2} + 16 y^{2} - 50 x + 160 y = - 25$ .

$25 {(x - 1)}^{2} - 25 + 16 y^{2} + 160 y = - 25$

Move $- 25$ to the right side of the equation by adding $25$ to both sides.

$25 {(x - 1)}^{2} + 16 y^{2} + 160 y = - 25 + 25$

Complete the square for $16 y^{2} + 160 y$ .

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

$a = 16$

$b = 160$

$c = 0$

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

$d = \frac{160}{2 \cdot 16}$

Simplify the right side.

Cancel the common factor of $160$ and $2$ .

Factor $2$ out of $160$ .

$d = \frac{2 \cdot 80}{2 \cdot 16}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 16$ .

$d = \frac{2 \cdot 80}{2 (16)}$

Cancel the common factor.

$d = \frac{2 \cdot 80}{2 \cdot 16}$

Rewrite the expression.

$d = \frac{80}{16}$

Cancel the common factor of $80$ and $16$ .

Factor $16$ out of $80$ .

$d = \frac{16 \cdot 5}{16}$

Cancel the common factors.

Factor $16$ out of $16$ .

$d = \frac{16 \cdot 5}{16 (1)}$

Cancel the common factor.

$d = \frac{16 \cdot 5}{16 \cdot 1}$

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

$e = 0 - \frac{160^{2}}{4 \cdot 16}$

Simplify the right side.

Simplify each term.

Raise $160$ to the power of $2$ .

$e = 0 - \frac{25600}{4 \cdot 16}$

Multiply $4$ by $16$ .

$e = 0 - \frac{25600}{64}$

Divide $25600$ by $64$ .

$e = 0 - 1 \cdot 400$

Multiply $- 1$ by $400$ .

$e = 0 - 400$

Subtract $400$ from $0$ .

$e = - 400$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $16 {(y + 5)}^{2} - 400$ .

$16 {(y + 5)}^{2} - 400$

Substitute $16 {(y + 5)}^{2} - 400$ for $16 y^{2} + 160 y$ in the equation $25 x^{2} + 16 y^{2} - 50 x + 160 y = - 25$ .

$25 {(x - 1)}^{2} + 16 {(y + 5)}^{2} - 400 = - 25 + 25$

Move $- 400$ to the right side of the equation by adding $400$ to both sides.

$25 {(x - 1)}^{2} + 16 {(y + 5)}^{2} = - 25 + 25 + 400$

Simplify $- 25 + 25 + 400$ .

Add $- 25$ and $25$ .

$25 {(x - 1)}^{2} + 16 {(y + 5)}^{2} = 0 + 400$

Add $0$ and $400$ .

$25 {(x - 1)}^{2} + 16 {(y + 5)}^{2} = 400$

Divide each term by $400$ to make the right side equal to one.

$\frac{25 {(x - 1)}^{2}}{400} + \frac{16 {(y + 5)}^{2}}{400} = \frac{400}{400}$

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 $1$ .

$\frac{{(x - 1)}^{2}}{16} + \frac{{(y + 5)}^{2}}{25} = 1$

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.

$a = 5$

$b = 4$

$k = - 5$

$h = 1$

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.

$(1, - 5 + 5)$

Simplify.

$(1, 0)$

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.

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

Simplify.

$(1, - 10)$

Ellipses have two vertices.

${Vertex}_{1}$ : $(1, 0)$

${Vertex}_{2}$ : $(1, - 10)$

${Vertex}_{1}$ : $(1, 0)$

${Vertex}_{2}$ : $(1, - 10)$

Step 5

Find the Vertices 25x^2+16y^2-50x+160y+25=0

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