Trigonometry Examples

$9 x^{2} + 16 y^{2} - 36 x + 160 y + 292 = 0$

Step 1

Find the standard form of the ellipse.

Tap for more steps...

Step 1.1

Subtract $292$ from both sides of the equation.

$9 x^{2} + 16 y^{2} - 36 x + 160 y = - 292$

Step 1.2

Complete the square for $9 x^{2} - 36 x$ .

Tap for more steps...

Step 1.2.1

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

$a = 9$

$b = - 36$

$c = 0$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Tap for more steps...

Step 1.2.3.1

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

$d = \frac{- 36}{2 \cdot 9}$

Step 1.2.3.2

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.1

Cancel the common factor of $- 36$ and $2$ .

Tap for more steps...

Step 1.2.3.2.1.1

Factor $2$ out of $- 36$ .

$d = \frac{2 \cdot - 18}{2 \cdot 9}$

Step 1.2.3.2.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.2.1.2.1

Factor $2$ out of $2 \cdot 9$ .

$d = \frac{2 \cdot - 18}{2 (9)}$

Step 1.2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 18}{2 \cdot 9}$

Step 1.2.3.2.1.2.3

Rewrite the expression.

$d = \frac{- 18}{9}$

Step 1.2.3.2.2

Cancel the common factor of $- 18$ and $9$ .

Tap for more steps...

Step 1.2.3.2.2.1

Factor $9$ out of $- 18$ .

$d = \frac{9 \cdot - 2}{9}$

Step 1.2.3.2.2.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.2.2.2.1

Factor $9$ out of $9$ .

$d = \frac{9 \cdot - 2}{9 (1)}$

Step 1.2.3.2.2.2.2

Cancel the common factor.

$d = \frac{9 \cdot - 2}{9 \cdot 1}$

Step 1.2.3.2.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.2.2.4

Divide $- 2$ by $1$ .

$d = - 2$

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 36)}^{2}}{4 \cdot 9}$

Step 1.2.4.2

Simplify the right side.

Tap for more steps...

Step 1.2.4.2.1

Simplify each term.

Tap for more steps...

Step 1.2.4.2.1.1

Raise $- 36$ to the power of $2$ .

$e = 0 - \frac{1296}{4 \cdot 9}$

Step 1.2.4.2.1.2

Multiply $4$ by $9$ .

$e = 0 - \frac{1296}{36}$

Step 1.2.4.2.1.3

Divide $1296$ by $36$ .

$e = 0 - 1 \cdot 36$

Step 1.2.4.2.1.4

Multiply $- 1$ by $36$ .

$e = 0 - 36$

Step 1.2.4.2.2

Subtract $36$ from $0$ .

$e = - 36$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(x - 2)}^{2} - 36$ .

$9 {(x - 2)}^{2} - 36$

Step 1.3

Substitute $9 {(x - 2)}^{2} - 36$ for $9 x^{2} - 36 x$ in the equation $9 x^{2} + 16 y^{2} - 36 x + 160 y = - 292$ .

$9 {(x - 2)}^{2} - 36 + 16 y^{2} + 160 y = - 292$

Step 1.4

Move $- 36$ to the right side of the equation by adding $36$ to both sides.

$9 {(x - 2)}^{2} + 16 y^{2} + 160 y = - 292 + 36$

Step 1.5

Complete the square for $16 y^{2} + 160 y$ .

Tap for more steps...

Step 1.5.1

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

$a = 16$

$b = 160$

$c = 0$

Step 1.5.2

Consider the vertex form of a parabola.

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

Step 1.5.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Tap for more steps...

Step 1.5.3.1

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

$d = \frac{160}{2 \cdot 16}$

Step 1.5.3.2

Simplify the right side.

Tap for more steps...

Step 1.5.3.2.1

Cancel the common factor of $160$ and $2$ .

Tap for more steps...

Step 1.5.3.2.1.1

Factor $2$ out of $160$ .

$d = \frac{2 \cdot 80}{2 \cdot 16}$

Step 1.5.3.2.1.2

Cancel the common factors.

Tap for more steps...

Step 1.5.3.2.1.2.1

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

$d = \frac{2 \cdot 80}{2 (16)}$

Step 1.5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 80}{2 \cdot 16}$

Step 1.5.3.2.1.2.3

Rewrite the expression.

$d = \frac{80}{16}$

Step 1.5.3.2.2

Cancel the common factor of $80$ and $16$ .

Tap for more steps...

Step 1.5.3.2.2.1

Factor $16$ out of $80$ .

$d = \frac{16 \cdot 5}{16}$

Step 1.5.3.2.2.2

Cancel the common factors.

Tap for more steps...

Step 1.5.3.2.2.2.1

Factor $16$ out of $16$ .

$d = \frac{16 \cdot 5}{16 (1)}$

Step 1.5.3.2.2.2.2

Cancel the common factor.

$d = \frac{16 \cdot 5}{16 \cdot 1}$

Step 1.5.3.2.2.2.3

Rewrite the expression.

$d = \frac{5}{1}$

Step 1.5.3.2.2.2.4

Divide $5$ by $1$ .

$d = 5$

Step 1.5.4

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

Tap for more steps...

Step 1.5.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{160^{2}}{4 \cdot 16}$

Step 1.5.4.2

Simplify the right side.

Tap for more steps...

Step 1.5.4.2.1

Simplify each term.

Tap for more steps...

Step 1.5.4.2.1.1

Raise $160$ to the power of $2$ .

$e = 0 - \frac{25600}{4 \cdot 16}$

Step 1.5.4.2.1.2

Multiply $4$ by $16$ .

$e = 0 - \frac{25600}{64}$

Step 1.5.4.2.1.3

Divide $25600$ by $64$ .

$e = 0 - 1 \cdot 400$

Step 1.5.4.2.1.4

Multiply $- 1$ by $400$ .

$e = 0 - 400$

Step 1.5.4.2.2

Subtract $400$ from $0$ .

$e = - 400$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $16 {(y + 5)}^{2} - 400$ .

$16 {(y + 5)}^{2} - 400$

Step 1.6

Substitute $16 {(y + 5)}^{2} - 400$ for $16 y^{2} + 160 y$ in the equation $9 x^{2} + 16 y^{2} - 36 x + 160 y = - 292$ .

$9 {(x - 2)}^{2} + 16 {(y + 5)}^{2} - 400 = - 292 + 36$

Step 1.7

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

$9 {(x - 2)}^{2} + 16 {(y + 5)}^{2} = - 292 + 36 + 400$

Step 1.8

Simplify $- 292 + 36 + 400$ .

Tap for more steps...

Step 1.8.1

Add $- 292$ and $36$ .

$9 {(x - 2)}^{2} + 16 {(y + 5)}^{2} = - 256 + 400$

Step 1.8.2

Add $- 256$ and $400$ .

$9 {(x - 2)}^{2} + 16 {(y + 5)}^{2} = 144$

Step 1.9

Divide each term by $144$ to make the right side equal to one.

$\frac{9 {(x - 2)}^{2}}{144} + \frac{16 {(y + 5)}^{2}}{144} = \frac{144}{144}$

Step 1.10

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 $1$ .

$\frac{{(x - 2)}^{2}}{16} + \frac{{(y + 5)}^{2}}{9} = 1$

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.

$a = 4$

$b = 3$

$k = - 5$

$h = 2$

Step 4

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

$(2, - 5)$

Step 5

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

Tap for more steps...

Step 5.1

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

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

Step 5.2

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

$\sqrt{{(4)}^{2} - {(3)}^{2}}$

Step 5.3

Simplify.

Tap for more steps...

Step 5.3.1

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

$\sqrt{16 - {(3)}^{2}}$

Step 5.3.2

Raise $3$ to the power of $2$ .

$\sqrt{16 - 1 \cdot 9}$

Step 5.3.3

Multiply $- 1$ by $9$ .

$\sqrt{16 - 9}$

Step 5.3.4

Subtract $9$ from $16$ .

$\sqrt{7}$

Step 6

Find the vertices.

Tap for more steps...

Step 6.1

The first vertex of an ellipse can be found by adding $a$ to $h$ .

$(h + a, k)$

Step 6.2

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

$(2 + 4, - 5)$

Step 6.3

Simplify.

$(6, - 5)$

Step 6.4

The second vertex of an ellipse can be found by subtracting $a$ from $h$ .

$(h - a, k)$

Step 6.5

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

$(2 - (4), - 5)$

Step 6.6

Simplify.

$(- 2, - 5)$

Step 6.7

Ellipses have two vertices.

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

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

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

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

Step 7

Find the foci.

Tap for more steps...

Step 7.1

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

$(h + c, k)$

Step 7.2

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

$(2 + \sqrt{7}, - 5)$

Step 7.3

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

$(h - c, k)$

Step 7.4

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

$(2 - (\sqrt{7}), - 5)$

Step 7.5

Simplify.

$(2 - \sqrt{7}, - 5)$

Step 7.6

Ellipses have two foci.

${Focus}_{1}$ : $(2 + \sqrt{7}, - 5)$

${Focus}_{2}$ : $(2 - \sqrt{7}, - 5)$

${Focus}_{1}$ : $(2 + \sqrt{7}, - 5)$

${Focus}_{2}$ : $(2 - \sqrt{7}, - 5)$

Step 8

Find the eccentricity.

Tap for more steps...

Step 8.1

Find the eccentricity by using the following formula.

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

Step 8.2

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

$\frac{\sqrt{{(4)}^{2} - {(3)}^{2}}}{4}$

Step 8.3

Simplify the numerator.

Tap for more steps...

Step 8.3.1

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

$\frac{\sqrt{16 - 3^{2}}}{4}$

Step 8.3.2

Raise $3$ to the power of $2$ .

$\frac{\sqrt{16 - 1 \cdot 9}}{4}$

Step 8.3.3

Multiply $- 1$ by $9$ .

$\frac{\sqrt{16 - 9}}{4}$

Step 8.3.4

Subtract $9$ from $16$ .

$\frac{\sqrt{7}}{4}$

Step 9

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

Center: $(2, - 5)$

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

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

${Focus}_{1}$ : $(2 + \sqrt{7}, - 5)$

${Focus}_{2}$ : $(2 - \sqrt{7}, - 5)$

Eccentricity: $\frac{\sqrt{7}}{4}$

Step 10