Trigonometry Examples

$16 x^{2} + 4 y^{2} + 96 x - 8 y + 84 = 0$

Step 1

Find the standard form of the ellipse.

Tap for more steps...

Step 1.1

Subtract $84$ from both sides of the equation.

$16 x^{2} + 4 y^{2} + 96 x - 8 y = - 84$

Step 1.2

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

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 $c$ .

$a = 16$

$b = 96$

$c = 0$

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

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

Step 1.2.3.2

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.1

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

Tap for more steps...

Step 1.2.3.2.1.1

Factor $2$ out of $96$ .

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

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 $2 \cdot 16$ .

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

Step 1.2.3.2.1.2.2

Cancel the common factor.

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

Step 1.2.3.2.1.2.3

Rewrite the expression.

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

Step 1.2.3.2.2

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

Tap for more steps...

Step 1.2.3.2.2.1

Factor $16$ out of $48$ .

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

Step 1.2.3.2.2.2

Cancel the common factors.

Tap for more steps...

Step 1.2.3.2.2.2.1

Factor $16$ out of $16$ .

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

Step 1.2.3.2.2.2.2

Cancel the common factor.

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

Step 1.2.3.2.2.2.3

Rewrite the expression.

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

Step 1.2.3.2.2.2.4

Divide $3$ by $1$ .

$d = 3$

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

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

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 $96$ to the power of $2$ .

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

Step 1.2.4.2.1.2

Multiply $4$ by $16$ .

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

Step 1.2.4.2.1.3

Divide $9216$ by $64$ .

$e = 0 - 1 \cdot 144$

Step 1.2.4.2.1.4

Multiply $- 1$ by $144$ .

$e = 0 - 144$

Step 1.2.4.2.2

Subtract $144$ from $0$ .

$e = - 144$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $16 {(x + 3)}^{2} - 144$ .

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

Step 1.3

Substitute $16 {(x + 3)}^{2} - 144$ for $16 x^{2} + 96 x$ in the equation $16 x^{2} + 4 y^{2} + 96 x - 8 y = - 84$ .

$16 {(x + 3)}^{2} - 144 + 4 y^{2} - 8 y = - 84$

Step 1.4

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

$16 {(x + 3)}^{2} + 4 y^{2} - 8 y = - 84 + 144$

Step 1.5

Complete the square for $4 y^{2} - 8 y$ .

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 $c$ .

$a = 4$

$b = - 8$

$c = 0$

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

$d = \frac{- 8}{2 \cdot 4}$

Step 1.5.3.2

Simplify the right side.

Tap for more steps...

Step 1.5.3.2.1

Cancel the common factor of $- 8$ and $2$ .

Tap for more steps...

Step 1.5.3.2.1.1

Factor $2$ out of $- 8$ .

$d = \frac{2 \cdot - 4}{2 \cdot 4}$

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 $2 \cdot 4$ .

$d = \frac{2 \cdot - 4}{2 (4)}$

Step 1.5.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 4}{2 \cdot 4}$

Step 1.5.3.2.1.2.3

Rewrite the expression.

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

Step 1.5.3.2.2

Cancel the common factor of $- 4$ and $4$ .

Tap for more steps...

Step 1.5.3.2.2.1

Factor $4$ out of $- 4$ .

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

Step 1.5.3.2.2.2

Cancel the common factors.

Tap for more steps...

Step 1.5.3.2.2.2.1

Factor $4$ out of $4$ .

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

Step 1.5.3.2.2.2.2

Cancel the common factor.

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

Step 1.5.3.2.2.2.3

Rewrite the expression.

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

Step 1.5.3.2.2.2.4

Divide $- 1$ by $1$ .

$d = - 1$

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

$e = 0 - \frac{{(- 8)}^{2}}{4 \cdot 4}$

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 $- 8$ to the power of $2$ .

$e = 0 - \frac{64}{4 \cdot 4}$

Step 1.5.4.2.1.2

Multiply $4$ by $4$ .

$e = 0 - \frac{64}{16}$

Step 1.5.4.2.1.3

Divide $64$ by $16$ .

$e = 0 - 1 \cdot 4$

Step 1.5.4.2.1.4

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Step 1.5.4.2.2

Subtract $4$ from $0$ .

$e = - 4$

Step 1.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(y - 1)}^{2} - 4$ .

$4 {(y - 1)}^{2} - 4$

Step 1.6

Substitute $4 {(y - 1)}^{2} - 4$ for $4 y^{2} - 8 y$ in the equation $16 x^{2} + 4 y^{2} + 96 x - 8 y = - 84$ .

$16 {(x + 3)}^{2} + 4 {(y - 1)}^{2} - 4 = - 84 + 144$

Step 1.7

Move $- 4$ to the right side of the equation by adding $4$ to both sides.

$16 {(x + 3)}^{2} + 4 {(y - 1)}^{2} = - 84 + 144 + 4$

Step 1.8

Simplify $- 84 + 144 + 4$ .

Tap for more steps...

Step 1.8.1

Add $- 84$ and $144$ .

$16 {(x + 3)}^{2} + 4 {(y - 1)}^{2} = 60 + 4$

Step 1.8.2

Add $60$ and $4$ .

$16 {(x + 3)}^{2} + 4 {(y - 1)}^{2} = 64$

Step 1.9

Divide each term by $64$ to make the right side equal to one.

$\frac{16 {(x + 3)}^{2}}{64} + \frac{4 {(y - 1)}^{2}}{64} = \frac{64}{64}$

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 $1$ .

$\frac{{(x + 3)}^{2}}{4} + \frac{{(y - 1)}^{2}}{16} = 1$

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.

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

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.

$a = 4$

$b = 2$

$k = 1$

$h = - 3$

Step 4

Find the vertices.

Tap for more steps...

Step 4.1

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

$(h, k + a)$

Step 4.2

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

$(- 3, 1 + 4)$

Step 4.3

Simplify.

$(- 3, 5)$

Step 4.4

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

$(h, k - a)$

Step 4.5

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

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

Step 4.6

Simplify.

$(- 3, - 3)$

Step 4.7

Ellipses have two vertices.

${Vertex}_{1}$ : $(- 3, 5)$

${Vertex}_{2}$ : $(- 3, - 3)$

${Vertex}_{1}$ : $(- 3, 5)$

${Vertex}_{2}$ : $(- 3, - 3)$

Step 5