Find the Vertices 16x^2+25y^2-32x+250y+241=0

$16 x^{2} + 25 y^{2} - 32 x + 250 y + 241 = 0$

Step 1

Find the standard form of the ellipse.

Subtract $241$ from both sides of the equation.

$16 x^{2} + 25 y^{2} - 32 x + 250 y = - 241$

Complete the square for $16 x^{2} - 32 x$ .

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

$a = 16$

$b = - 32$

$c = 0$

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

$d = \frac{- 32}{2 \cdot 16}$

Simplify the right side.

Cancel the common factor of $- 32$ and $2$ .

Factor $2$ out of $- 32$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor $16$ out of $- 16$ .

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

Cancel the common factors.

Factor $16$ out of $16$ .

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

Cancel the common factor.

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

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

$e = 0 - \frac{{(- 32)}^{2}}{4 \cdot 16}$

Simplify the right side.

Simplify each term.

Raise $- 32$ to the power of $2$ .

$e = 0 - \frac{1024}{4 \cdot 16}$

Multiply $4$ by $16$ .

$e = 0 - \frac{1024}{64}$

Divide $1024$ by $64$ .

$e = 0 - 1 \cdot 16$

Multiply $- 1$ by $16$ .

$e = 0 - 16$

Subtract $16$ from $0$ .

$e = - 16$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $16 {(x - 1)}^{2} - 16$ .

$16 {(x - 1)}^{2} - 16$

Substitute $16 {(x - 1)}^{2} - 16$ for $16 x^{2} - 32 x$ in the equation $16 x^{2} + 25 y^{2} - 32 x + 250 y = - 241$ .

$16 {(x - 1)}^{2} - 16 + 25 y^{2} + 250 y = - 241$

Move $- 16$ to the right side of the equation by adding $16$ to both sides.

$16 {(x - 1)}^{2} + 25 y^{2} + 250 y = - 241 + 16$

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 $16 x^{2} + 25 y^{2} - 32 x + 250 y = - 241$ .

$16 {(x - 1)}^{2} + 25 {(y + 5)}^{2} - 625 = - 241 + 16$

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

$16 {(x - 1)}^{2} + 25 {(y + 5)}^{2} = - 241 + 16 + 625$

Simplify $- 241 + 16 + 625$ .

Add $- 241$ and $16$ .

$16 {(x - 1)}^{2} + 25 {(y + 5)}^{2} = - 225 + 625$

Add $- 225$ and $625$ .

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

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

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

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 $1$ .

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

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.

$a = 5$

$b = 4$

$k = - 5$

$h = 1$

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.

$(1 + 5, - 5)$

Simplify.

$(6, - 5)$

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.

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

Simplify.

$(- 4, - 5)$

Ellipses have two vertices.

${Vertex}_{1}$ : $(6, - 5)$

${Vertex}_{2}$ : $(- 4, - 5)$

${Vertex}_{1}$ : $(6, - 5)$

${Vertex}_{2}$ : $(- 4, - 5)$

Step 5

Find the Vertices 16x^2+25y^2-32x+250y+241=0

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