Find the Axis of Symmetry 9x^2+4y^2-90x-24y+225=0

$9 x^{2} + 4 y^{2} - 90 x - 24 y + 225 = 0$

Step 1

Find the standard form of the ellipse.

Subtract $225$ from both sides of the equation.

$9 x^{2} + 4 y^{2} - 90 x - 24 y = - 225$

Complete the square for $9 x^{2} - 90 x$ .

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

$a = 9$

$b = - 90$

$c = 0$

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

$d = \frac{- 90}{2 \cdot 9}$

Simplify the right side.

Cancel the common factor of $- 90$ and $2$ .

Factor $2$ out of $- 90$ .

$d = \frac{2 \cdot - 45}{2 \cdot 9}$

Cancel the common factors.

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

$d = \frac{2 \cdot - 45}{2 (9)}$

Cancel the common factor.

$d = \frac{2 \cdot - 45}{2 \cdot 9}$

Rewrite the expression.

$d = \frac{- 45}{9}$

Cancel the common factor of $- 45$ and $9$ .

Factor $9$ out of $- 45$ .

$d = \frac{9 \cdot - 5}{9}$

Cancel the common factors.

Factor $9$ out of $9$ .

$d = \frac{9 \cdot - 5}{9 (1)}$

Cancel the common factor.

$d = \frac{9 \cdot - 5}{9 \cdot 1}$

Rewrite the expression.

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

Divide $- 5$ by $1$ .

$d = - 5$

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

$e = 0 - \frac{{(- 90)}^{2}}{4 \cdot 9}$

Simplify the right side.

Simplify each term.

Raise $- 90$ to the power of $2$ .

$e = 0 - \frac{8100}{4 \cdot 9}$

Multiply $4$ by $9$ .

$e = 0 - \frac{8100}{36}$

Divide $8100$ by $36$ .

$e = 0 - 1 \cdot 225$

Multiply $- 1$ by $225$ .

$e = 0 - 225$

Subtract $225$ from $0$ .

$e = - 225$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $9 {(x - 5)}^{2} - 225$ .

$9 {(x - 5)}^{2} - 225$

Substitute $9 {(x - 5)}^{2} - 225$ for $9 x^{2} - 90 x$ in the equation $9 x^{2} + 4 y^{2} - 90 x - 24 y = - 225$ .

$9 {(x - 5)}^{2} - 225 + 4 y^{2} - 24 y = - 225$

Move $- 225$ to the right side of the equation by adding $225$ to both sides.

$9 {(x - 5)}^{2} + 4 y^{2} - 24 y = - 225 + 225$

Complete the square for $4 y^{2} - 24 y$ .

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

$a = 4$

$b = - 24$

$c = 0$

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

$d = \frac{- 24}{2 \cdot 4}$

Simplify the right side.

Cancel the common factor of $- 24$ and $2$ .

Factor $2$ out of $- 24$ .

$d = \frac{2 \cdot - 12}{2 \cdot 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 4$ .

$d = \frac{2 \cdot - 12}{2 (4)}$

Cancel the common factor.

$d = \frac{2 \cdot - 12}{2 \cdot 4}$

Rewrite the expression.

$d = \frac{- 12}{4}$

Cancel the common factor of $- 12$ and $4$ .

Factor $4$ out of $- 12$ .

$d = \frac{4 \cdot - 3}{4}$

Cancel the common factors.

Factor $4$ out of $4$ .

$d = \frac{4 \cdot - 3}{4 (1)}$

Cancel the common factor.

$d = \frac{4 \cdot - 3}{4 \cdot 1}$

Rewrite the expression.

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

Divide $- 3$ by $1$ .

$d = - 3$

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

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

$e = 0 - \frac{{(- 24)}^{2}}{4 \cdot 4}$

Simplify the right side.

Simplify each term.

Raise $- 24$ to the power of $2$ .

$e = 0 - \frac{576}{4 \cdot 4}$

Multiply $4$ by $4$ .

$e = 0 - \frac{576}{16}$

Divide $576$ by $16$ .

$e = 0 - 1 \cdot 36$

Multiply $- 1$ by $36$ .

$e = 0 - 36$

Subtract $36$ from $0$ .

$e = - 36$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(y - 3)}^{2} - 36$ .

$4 {(y - 3)}^{2} - 36$

Substitute $4 {(y - 3)}^{2} - 36$ for $4 y^{2} - 24 y$ in the equation $9 x^{2} + 4 y^{2} - 90 x - 24 y = - 225$ .

$9 {(x - 5)}^{2} + 4 {(y - 3)}^{2} - 36 = - 225 + 225$

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

$9 {(x - 5)}^{2} + 4 {(y - 3)}^{2} = - 225 + 225 + 36$

Simplify $- 225 + 225 + 36$ .

Add $- 225$ and $225$ .

$9 {(x - 5)}^{2} + 4 {(y - 3)}^{2} = 0 + 36$

Add $0$ and $36$ .

$9 {(x - 5)}^{2} + 4 {(y - 3)}^{2} = 36$

Divide each term by $36$ to make the right side equal to one.

$\frac{9 {(x - 5)}^{2}}{36} + \frac{4 {(y - 3)}^{2}}{36} = \frac{36}{36}$

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 $1$ .

$\frac{{(x - 5)}^{2}}{4} + \frac{{(y - 3)}^{2}}{9} = 1$

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.

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

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.

$a = 3$

$b = 2$

$k = 3$

$h = 5$

Step 4

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

$(5, 3)$

Step 5

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.

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

Simplify.

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

$\sqrt{9 - {(2)}^{2}}$

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

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

Multiply $- 1$ by $4$ .

$\sqrt{9 - 4}$

Subtract $4$ from $9$ .

$\sqrt{5}$

Step 6

Find the vertices.

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.

$(5, 3 + 3)$

Simplify.

$(5, 6)$

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.

$(5, 3 - (3))$

Simplify.

$(5, 0)$

Ellipses have two vertices.

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

${Vertex}_{2}$ : $(5, 0)$

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

${Vertex}_{2}$ : $(5, 0)$

Step 7

Find the foci.

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.

$(5, 3 + \sqrt{5})$

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.

$(5, 3 - (\sqrt{5}))$

Simplify.

$(5, 3 - \sqrt{5})$

Ellipses have two foci.

${Focus}_{1}$ : $(5, 3 + \sqrt{5})$

${Focus}_{2}$ : $(5, 3 - \sqrt{5})$

${Focus}_{1}$ : $(5, 3 + \sqrt{5})$

${Focus}_{2}$ : $(5, 3 - \sqrt{5})$

Step 8

Find the eccentricity.

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.

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

Simplify the numerator.

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

$\frac{\sqrt{9 - 2^{2}}}{3}$

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

$\frac{\sqrt{9 - 1 \cdot 4}}{3}$

Multiply $- 1$ by $4$ .

$\frac{\sqrt{9 - 4}}{3}$

Subtract $4$ from $9$ .

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

Step 9

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

Center: $(5, 3)$

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

${Vertex}_{2}$ : $(5, 0)$

${Focus}_{1}$ : $(5, 3 + \sqrt{5})$

${Focus}_{2}$ : $(5, 3 - \sqrt{5})$

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

Step 10

Find the Axis of Symmetry 9x^2+4y^2-90x-24y+225=0

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