Find the Foci 16x^2+25y^2-128x+50y-119=0

$16 x^{2} + 25 y^{2} - 128 x + 50 y - 119 = 0$

Step 1

Find the standard form of the ellipse.

Add $119$ to both sides of the equation.

$16 x^{2} + 25 y^{2} - 128 x + 50 y = 119$

Complete the square for $16 x^{2} - 128 x$ .

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

$a = 16$

$b = - 128$

$c = 0$

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

$d = \frac{- 128}{2 \cdot 16}$

Simplify the right side.

Cancel the common factor of $- 128$ and $2$ .

Factor $2$ out of $- 128$ .

$d = \frac{2 \cdot - 64}{2 \cdot 16}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 64}{2 (16)}$

Cancel the common factor.

$d = \frac{2 \cdot - 64}{2 \cdot 16}$

Rewrite the expression.

$d = \frac{- 64}{16}$

Cancel the common factor of $- 64$ and $16$ .

Factor $16$ out of $- 64$ .

$d = \frac{16 \cdot - 4}{16}$

Cancel the common factors.

Factor $16$ out of $16$ .

$d = \frac{16 \cdot - 4}{16 (1)}$

Cancel the common factor.

$d = \frac{16 \cdot - 4}{16 \cdot 1}$

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

$e = 0 - \frac{{(- 128)}^{2}}{4 \cdot 16}$

Simplify the right side.

Simplify each term.

Raise $- 128$ to the power of $2$ .

$e = 0 - \frac{16384}{4 \cdot 16}$

Multiply $4$ by $16$ .

$e = 0 - \frac{16384}{64}$

Divide $16384$ by $64$ .

$e = 0 - 1 \cdot 256$

Multiply $- 1$ by $256$ .

$e = 0 - 256$

Subtract $256$ from $0$ .

$e = - 256$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $16 {(x - 4)}^{2} - 256$ .

$16 {(x - 4)}^{2} - 256$

Substitute $16 {(x - 4)}^{2} - 256$ for $16 x^{2} - 128 x$ in the equation $16 x^{2} + 25 y^{2} - 128 x + 50 y = 119$ .

$16 {(x - 4)}^{2} - 256 + 25 y^{2} + 50 y = 119$

Move $- 256$ to the right side of the equation by adding $256$ to both sides.

$16 {(x - 4)}^{2} + 25 y^{2} + 50 y = 119 + 256$

Complete the square for $25 y^{2} + 50 y$ .

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 $25$ .

Cancel the common factor.

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

Rewrite the expression.

$d = 1$

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 $25 {(y + 1)}^{2} - 25$ .

$25 {(y + 1)}^{2} - 25$

Substitute $25 {(y + 1)}^{2} - 25$ for $25 y^{2} + 50 y$ in the equation $16 x^{2} + 25 y^{2} - 128 x + 50 y = 119$ .

$16 {(x - 4)}^{2} + 25 {(y + 1)}^{2} - 25 = 119 + 256$

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

$16 {(x - 4)}^{2} + 25 {(y + 1)}^{2} = 119 + 256 + 25$

Simplify $119 + 256 + 25$ .

Add $119$ and $256$ .

$16 {(x - 4)}^{2} + 25 {(y + 1)}^{2} = 375 + 25$

Add $375$ and $25$ .

$16 {(x - 4)}^{2} + 25 {(y + 1)}^{2} = 400$

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

$\frac{16 {(x - 4)}^{2}}{400} + \frac{25 {(y + 1)}^{2}}{400} = \frac{400}{400}$

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 $1$ .

$\frac{{(x - 4)}^{2}}{25} + \frac{{(y + 1)}^{2}}{16} = 1$

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.

$a = 5$

$b = 4$

$k = - 1$

$h = 4$

Step 4

Find $c$ , the distance from the center to a focus.

Find the distance from the center to a focus of the ellipse by using the following formula.

$\sqrt{a^{2} - b^{2}}$

Substitute the values of $a$ and $b$ in the formula.

$\sqrt{{(5)}^{2} - {(4)}^{2}}$

Simplify.

Raise $5$ to the power of $2$ .

$\sqrt{25 - {(4)}^{2}}$

Raise $4$ to the power of $2$ .

$\sqrt{25 - 1 \cdot 16}$

Multiply $- 1$ by $16$ .

$\sqrt{25 - 16}$

Subtract $16$ from $25$ .

$\sqrt{9}$

Rewrite $9$ as $3^{2}$ .

$\sqrt{3^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$3$

Step 5

Find the foci.

The first focus of an ellipse can be found by adding $c$ to $h$ .

$(h + c, k)$

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

$(4 + 3, - 1)$

Simplify.

$(7, - 1)$

The second focus of an ellipse can be found by subtracting $c$ from $h$ .

$(h - c, k)$

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

$(4 - (3), - 1)$

Simplify.

$(1, - 1)$

Ellipses have two foci.

${Focus}_{1}$ : $(7, - 1)$

${Focus}_{2}$ : $(1, - 1)$

${Focus}_{1}$ : $(7, - 1)$

${Focus}_{2}$ : $(1, - 1)$

Step 6

Find the Foci 16x^2+25y^2-128x+50y-119=0

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