Graph 25x^2+16y^2-100x+96y-156=0

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Trigonometry

$25 x^{2} + 16 y^{2} - 100 x + 96 y - 156 = 0$

Find the standard form of the ellipse.

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Add $156$ to both sides of the equation.

$25 x^{2} + 16 y^{2} - 100 x + 96 y = 156$

Complete the square for $25 x^{2} - 100 x$ .

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

$a = 25, b = - 100, c = 0$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

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

Simplify the right side.

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Cancel the common factor of $100$ and $2$ .

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Factor $2$ out of $100$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $50$ and $25$ .

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Factor $25$ out of $50$ .

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

Cancel the common factors.

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Factor $25$ out of $25$ .

$d = - \frac{25 \cdot 2}{25 (1)}$

Cancel the common factor.

$d = - \frac{25 \cdot 2}{25 \cdot 1}$

Rewrite the expression.

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

Divide $2$ by $1$ .

$d = - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$d = - 2$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Simplify each term.

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Raise $- 100$ to the power of $2$ .

$e = 0 - \frac{10000}{4 (25)}$

Multiply $4$ by $25$ .

$e = 0 - \frac{10000}{100}$

Divide $10000$ by $100$ .

$e = 0 - 1 \cdot 100$

Multiply $- 1$ by $100$ .

$e = 0 - 100$

Subtract $100$ from $0$ .

$e = - 100$

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

$25 {(x - 2)}^{2} - 100$

Substitute $25 {(x - 2)}^{2} - 100$ for $25 x^{2} - 100 x$ in the equation $25 x^{2} + 16 y^{2} - 100 x + 96 y = 156$ .

$25 {(x - 2)}^{2} - 100 + 16 y^{2} + 96 y = 156$

Move $- 100$ to the right side of the equation by adding $100$ to both sides.

$25 {(x - 2)}^{2} + 16 y^{2} + 96 y = 156 + 100$

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

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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.

$a {(x + d)}^{2} + e$

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{96}{2 (16)}$

Simplify the right side.

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

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Factor $2$ out of $96$ .

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

Cancel the common factors.

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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 $16$ .

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Factor $16$ out of $48$ .

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

Cancel the common factors.

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

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Simplify each term.

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

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

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 $a {(x + d)}^{2} + e$ .

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

Substitute $16 {(y + 3)}^{2} - 144$ for $16 y^{2} + 96 y$ in the equation $25 x^{2} + 16 y^{2} - 100 x + 96 y = 156$ .

$25 {(x - 2)}^{2} + 16 {(y + 3)}^{2} - 144 = 156 + 100$

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

$25 {(x - 2)}^{2} + 16 {(y + 3)}^{2} = 156 + 100 + 144$

Simplify $156 + 100 + 144$ .

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Add $156$ and $100$ .

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

Add $256$ and $144$ .

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

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

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

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 $1$ .

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

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.

$\frac{{(x - h)}^{2}}{b^{2}} + \frac{{(y - k)}^{2}}{a^{2}} = 1$

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.

$a = 5$

$b = 4$

$k = - 3$

$h = 2$

The center of an ellipse follows the form of $(h, k)$ . Substitute in the values of $h$ and $k$ .

$(2, - 3)$

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

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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.

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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.

$3$

Find the vertices.

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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.

$(2, - 3 + 5)$

Simplify.

$(2, 2)$

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.

$(2, - 3 - (5))$

Simplify.

$(2, - 8)$

Ellipses have two vertices.

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

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

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

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

Find the foci.

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

$(h, k + c)$

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

$(2, - 3 + 3)$

Simplify.

$(2, 0)$

The first focus of an ellipse can be found by subtracting $c$ from $k$ .

$(h, k - c)$

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

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

Simplify.

$(2, - 6)$

Ellipses have two foci.

${Focus}_{1}$ : $(2, 0)$

${Focus}_{2}$ : $(2, - 6)$

${Focus}_{1}$ : $(2, 0)$

${Focus}_{2}$ : $(2, - 6)$

Find the eccentricity.

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Find the eccentricity by using the following formula.

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

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

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

Simplify the numerator.

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

$\frac{\sqrt{25 - 4^{2}}}{5}$

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

$\frac{\sqrt{25 - 1 \cdot 16}}{5}$

Multiply $- 1$ by $16$ .

$\frac{\sqrt{25 - 16}}{5}$

Subtract $16$ from $25$ .

$\frac{\sqrt{9}}{5}$

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

$\frac{\sqrt{3^{2}}}{5}$

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

$\frac{3}{5}$

These values represent the important values for graphing and analyzing an ellipse.

Center: $(2, - 3)$

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

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

${Focus}_{1}$ : $(2, 0)$

${Focus}_{2}$ : $(2, - 6)$

Eccentricity: $\frac{3}{5}$

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