Find the Foci 16x^2+25y^2-64x+200y+64=0

$16 x^{2} + 25 y^{2} - 64 x + 200 y + 64 = 0$

Step 1

Find the standard form of the ellipse.

Subtract $64$ from both sides of the equation.

$16 x^{2} + 25 y^{2} - 64 x + 200 y = - 64$

Complete the square for $16 x^{2} - 64 x$ .

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

$a = 16$

$b = - 64$

$c = 0$

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

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

Simplify the right side.

Cancel the common factor of $- 64$ and $2$ .

Factor $2$ out of $- 64$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $- 32$ and $16$ .

Factor $16$ out of $- 32$ .

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

Cancel the common factors.

Factor $16$ out of $16$ .

$d = \frac{16 \cdot - 2}{16 (1)}$

Cancel the common factor.

$d = \frac{16 \cdot - 2}{16 \cdot 1}$

Rewrite the expression.

$d = \frac{- 2}{1}$

Divide $- 2$ by $1$ .

$d = - 2$

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

$e = 0 - \frac{{(- 64)}^{2}}{4 \cdot 16}$

Simplify the right side.

Simplify each term.

Raise $- 64$ to the power of $2$ .

$e = 0 - \frac{4096}{4 \cdot 16}$

Multiply $4$ by $16$ .

$e = 0 - \frac{4096}{64}$

Divide $4096$ by $64$ .

$e = 0 - 1 \cdot 64$

Multiply $- 1$ by $64$ .

$e = 0 - 64$

Subtract $64$ from $0$ .

$e = - 64$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $16 {(x - 2)}^{2} - 64$ .

$16 {(x - 2)}^{2} - 64$

Substitute $16 {(x - 2)}^{2} - 64$ for $16 x^{2} - 64 x$ in the equation $16 x^{2} + 25 y^{2} - 64 x + 200 y = - 64$ .

$16 {(x - 2)}^{2} - 64 + 25 y^{2} + 200 y = - 64$

Move $- 64$ to the right side of the equation by adding $64$ to both sides.

$16 {(x - 2)}^{2} + 25 y^{2} + 200 y = - 64 + 64$

Complete the square for $25 y^{2} + 200 y$ .

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

$a = 25$

$b = 200$

$c = 0$

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

$d = \frac{200}{2 \cdot 25}$

Simplify the right side.

Cancel the common factor of $200$ and $2$ .

Factor $2$ out of $200$ .

$d = \frac{2 \cdot 100}{2 \cdot 25}$

Cancel the common factors.

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

$d = \frac{2 \cdot 100}{2 (25)}$

Cancel the common factor.

$d = \frac{2 \cdot 100}{2 \cdot 25}$

Rewrite the expression.

$d = \frac{100}{25}$

Cancel the common factor of $100$ and $25$ .

Factor $25$ out of $100$ .

$d = \frac{25 \cdot 4}{25}$

Cancel the common factors.

Factor $25$ out of $25$ .

$d = \frac{25 \cdot 4}{25 (1)}$

Cancel the common factor.

$d = \frac{25 \cdot 4}{25 \cdot 1}$

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

$e = 0 - \frac{200^{2}}{4 \cdot 25}$

Simplify the right side.

Simplify each term.

Raise $200$ to the power of $2$ .

$e = 0 - \frac{40000}{4 \cdot 25}$

Multiply $4$ by $25$ .

$e = 0 - \frac{40000}{100}$

Divide $40000$ by $100$ .

$e = 0 - 1 \cdot 400$

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 $25 {(y + 4)}^{2} - 400$ .

$25 {(y + 4)}^{2} - 400$

Substitute $25 {(y + 4)}^{2} - 400$ for $25 y^{2} + 200 y$ in the equation $16 x^{2} + 25 y^{2} - 64 x + 200 y = - 64$ .

$16 {(x - 2)}^{2} + 25 {(y + 4)}^{2} - 400 = - 64 + 64$

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

$16 {(x - 2)}^{2} + 25 {(y + 4)}^{2} = - 64 + 64 + 400$

Simplify $- 64 + 64 + 400$ .

Add $- 64$ and $64$ .

$16 {(x - 2)}^{2} + 25 {(y + 4)}^{2} = 0 + 400$

Add $0$ and $400$ .

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

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

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

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 $1$ .

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

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.

$a = 5$

$b = 4$

$k = - 4$

$h = 2$

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.

$(2 + 3, - 4)$

Simplify.

$(5, - 4)$

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.

$(2 - (3), - 4)$

Simplify.

$(- 1, - 4)$

Ellipses have two foci.

${Focus}_{1}$ : $(5, - 4)$

${Focus}_{2}$ : $(- 1, - 4)$

${Focus}_{1}$ : $(5, - 4)$

${Focus}_{2}$ : $(- 1, - 4)$

Step 6

Find the Foci 16x^2+25y^2-64x+200y+64=0

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