Find the Vertices 25x^2+16y^2-50x+96y-231=0

$25 x^{2} + 16 y^{2} - 50 x + 96 y - 231 = 0$

Step 1

Find the standard form of the ellipse.

Add $231$ to both sides of the equation.

$25 x^{2} + 16 y^{2} - 50 x + 96 y = 231$

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 $25 x^{2} + 16 y^{2} - 50 x + 96 y = 231$ .

$25 {(x - 1)}^{2} - 25 + 16 y^{2} + 96 y = 231$

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

$25 {(x - 1)}^{2} + 16 y^{2} + 96 y = 231 + 25$

Complete the square for $16 y^{2} + 96 y$ .

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

$a = 16$

$b = 96$

$c = 0$

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

$d = \frac{96}{2 \cdot 16}$

Simplify the right side.

Cancel the common factor of $96$ and $2$ .

Factor $2$ out of $96$ .

$d = \frac{2 \cdot 48}{2 \cdot 16}$

Cancel the common factors.

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

$d = \frac{2 \cdot 48}{2 (16)}$

Cancel the common factor.

$d = \frac{2 \cdot 48}{2 \cdot 16}$

Rewrite the expression.

$d = \frac{48}{16}$

Cancel the common factor of $48$ and $16$ .

Factor $16$ out of $48$ .

$d = \frac{16 \cdot 3}{16}$

Cancel the common factors.

Factor $16$ out of $16$ .

$d = \frac{16 \cdot 3}{16 (1)}$

Cancel the common factor.

$d = \frac{16 \cdot 3}{16 \cdot 1}$

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

$e = 0 - \frac{96^{2}}{4 \cdot 16}$

Simplify the right side.

Simplify each term.

Raise $96$ to the power of $2$ .

$e = 0 - \frac{9216}{4 \cdot 16}$

Multiply $4$ by $16$ .

$e = 0 - \frac{9216}{64}$

Divide $9216$ by $64$ .

$e = 0 - 1 \cdot 144$

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 $16 {(y + 3)}^{2} - 144$ .

$16 {(y + 3)}^{2} - 144$

Substitute $16 {(y + 3)}^{2} - 144$ for $16 y^{2} + 96 y$ in the equation $25 x^{2} + 16 y^{2} - 50 x + 96 y = 231$ .

$25 {(x - 1)}^{2} + 16 {(y + 3)}^{2} - 144 = 231 + 25$

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

$25 {(x - 1)}^{2} + 16 {(y + 3)}^{2} = 231 + 25 + 144$

Simplify $231 + 25 + 144$ .

Add $231$ and $25$ .

$25 {(x - 1)}^{2} + 16 {(y + 3)}^{2} = 256 + 144$

Add $256$ and $144$ .

$25 {(x - 1)}^{2} + 16 {(y + 3)}^{2} = 400$

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

$\frac{25 {(x - 1)}^{2}}{400} + \frac{16 {(y + 3)}^{2}}{400} = \frac{400}{400}$

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 $1$ .

$\frac{{(x - 1)}^{2}}{16} + \frac{{(y + 3)}^{2}}{25} = 1$

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.

$a = 5$

$b = 4$

$k = - 3$

$h = 1$

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.

$(1, - 3 + 5)$

Simplify.

$(1, 2)$

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.

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

Simplify.

$(1, - 8)$

Ellipses have two vertices.

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

${Vertex}_{2}$ : $(1, - 8)$

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

${Vertex}_{2}$ : $(1, - 8)$

Step 5

Find the Vertices 25x^2+16y^2-50x+96y-231=0

Related Questions

Support LCMGCF.com by disabling your adblocker.