Find the Axis of Symmetry 3x^2+y^2-6x-8y-5=0

$3 x^{2} + y^{2} - 6 x - 8 y - 5 = 0$

Find the standard form of the ellipse.

Add $5$ to both sides of the equation.

$3 x^{2} + y^{2} - 6 x - 8 y = 5$

Complete the square for $3 x^{2} - 6 x$ .

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

$a = 3$

$b = - 6$

$c = 0$

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

$d = \frac{- 6}{2 \cdot 3}$

Simplify the right side.

Cancel the common factor of $- 6$ and $2$ .

Factor $2$ out of $- 6$ .

$d = \frac{2 \cdot - 3}{2 \cdot 3}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 3$ .

$d = \frac{2 \cdot - 3}{2 (3)}$

Cancel the common factor.

$d = \frac{2 \cdot - 3}{2 \cdot 3}$

Rewrite the expression.

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

Cancel the common factor of $- 3$ and $3$ .

Factor $3$ out of $- 3$ .

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

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 1$ by $1$ .

$d = - 1$

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

$e = 0 - \frac{{(- 6)}^{2}}{4 \cdot 3}$

Simplify the right side.

Simplify each term.

Raise $- 6$ to the power of $2$ .

$e = 0 - \frac{36}{4 \cdot 3}$

Multiply $4$ by $3$ .

$e = 0 - \frac{36}{12}$

Divide $36$ by $12$ .

$e = 0 - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$e = 0 - 3$

Subtract $3$ from $0$ .

$e = - 3$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $3 {(x - 1)}^{2} - 3$ .

$3 {(x - 1)}^{2} - 3$

Substitute $3 {(x - 1)}^{2} - 3$ for $3 x^{2} - 6 x$ in the equation $3 x^{2} + y^{2} - 6 x - 8 y = 5$ .

$3 {(x - 1)}^{2} - 3 + y^{2} - 8 y = 5$

Move $- 3$ to the right side of the equation by adding $3$ to both sides.

$3 {(x - 1)}^{2} + y^{2} - 8 y = 5 + 3$

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

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

$a = 1$

$b = - 8$

$c = 0$

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

$d = \frac{- 8}{2 \cdot 1}$

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

Factor $2$ out of $- 8$ .

$d = \frac{2 \cdot - 4}{2 \cdot 1}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot - 4}{2 (1)}$

Cancel the common factor.

$d = \frac{2 \cdot - 4}{2 \cdot 1}$

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

$e = 0 - \frac{{(- 8)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

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

$e = 0 - \frac{64}{4 \cdot 1}$

Multiply $4$ by $1$ .

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

Divide $64$ by $4$ .

$e = 0 - 1 \cdot 16$

Multiply $- 1$ by $16$ .

$e = 0 - 16$

Subtract $16$ from $0$ .

$e = - 16$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - 4)}^{2} - 16$ .

${(y - 4)}^{2} - 16$

Substitute ${(y - 4)}^{2} - 16$ for $y^{2} - 8 y$ in the equation $3 x^{2} + y^{2} - 6 x - 8 y = 5$ .

$3 {(x - 1)}^{2} + {(y - 4)}^{2} - 16 = 5 + 3$

Move $- 16$ to the right side of the equation by adding $16$ to both sides.

$3 {(x - 1)}^{2} + {(y - 4)}^{2} = 5 + 3 + 16$

Simplify $5 + 3 + 16$ .

Add $5$ and $3$ .

$3 {(x - 1)}^{2} + {(y - 4)}^{2} = 8 + 16$

Add $8$ and $16$ .

$3 {(x - 1)}^{2} + {(y - 4)}^{2} = 24$

Divide each term by $24$ to make the right side equal to one.

$\frac{3 {(x - 1)}^{2}}{24} + \frac{{(y - 4)}^{2}}{24} = \frac{24}{24}$

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 $1$ .

$\frac{{(x - 1)}^{2}}{8} + \frac{{(y - 4)}^{2}}{24} = 1$

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.

$a = 2 \sqrt{6}$

$b = 2 \sqrt{2}$

$k = 4$

$h = 1$

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

$(1, 4)$

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.

$\sqrt{{(2 \sqrt{6})}^{2} - {(2 \sqrt{2})}^{2}}$

Simplify.

Simplify the expression.

Apply the product rule to $2 \sqrt{6}$ .

$\sqrt{2^{2} {\sqrt{6}}^{2} - {(2 \sqrt{2})}^{2}}$

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

$\sqrt{4 {\sqrt{6}}^{2} - {(2 \sqrt{2})}^{2}}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\sqrt{4 {(6^{\frac{1}{2}})}^{2} - {(2 \sqrt{2})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{4 \cdot 6^{\frac{1}{2} \cdot 2} - {(2 \sqrt{2})}^{2}}$

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{4 \cdot 6^{\frac{2}{2}} - {(2 \sqrt{2})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{4 \cdot 6^{\frac{2}{2}} - {(2 \sqrt{2})}^{2}}$

Rewrite the expression.

$\sqrt{4 \cdot 6^{1} - {(2 \sqrt{2})}^{2}}$

Evaluate the exponent.

$\sqrt{4 \cdot 6 - {(2 \sqrt{2})}^{2}}$

Simplify the expression.

Multiply $4$ by $6$ .

$\sqrt{24 - {(2 \sqrt{2})}^{2}}$

Apply the product rule to $2 \sqrt{2}$ .

$\sqrt{24 - (2^{2} {\sqrt{2}}^{2})}$

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

$\sqrt{24 - (4 {\sqrt{2}}^{2})}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sqrt{24 - (4 {(2^{\frac{1}{2}})}^{2})}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{24 - (4 \cdot 2^{\frac{1}{2} \cdot 2})}$

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{24 - (4 \cdot 2^{\frac{2}{2}})}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{24 - (4 \cdot 2^{\frac{2}{2}})}$

Rewrite the expression.

$\sqrt{24 - (4 \cdot 2^{1})}$

Evaluate the exponent.

$\sqrt{24 - (4 \cdot 2)}$

Simplify the expression.

Multiply $4$ by $2$ .

$\sqrt{24 - 1 \cdot 8}$

Multiply $- 1$ by $8$ .

$\sqrt{24 - 8}$

Subtract $8$ from $24$ .

$\sqrt{16}$

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2}}$

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

$4$

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.

$(1, 4 + 2 \sqrt{6})$

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.

$(1, 4 - (2 \sqrt{6}))$

Simplify.

$(1, 4 - 2 \sqrt{6})$

Ellipses have two vertices.

${Vertex}_{1}$ : $(1, 4 + 2 \sqrt{6})$

${Vertex}_{2}$ : $(1, 4 - 2 \sqrt{6})$

${Vertex}_{1}$ : $(1, 4 + 2 \sqrt{6})$

${Vertex}_{2}$ : $(1, 4 - 2 \sqrt{6})$

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.

$(1, 4 + 4)$

Simplify.

$(1, 8)$

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.

$(1, 4 - (4))$

Simplify.

$(1, 0)$

Ellipses have two foci.

${Focus}_{1}$ : $(1, 8)$

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

${Focus}_{1}$ : $(1, 8)$

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

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.

$\frac{\sqrt{{(2 \sqrt{6})}^{2} - {(2 \sqrt{2})}^{2}}}{2 \sqrt{6}}$

Simplify.

Simplify the numerator.

Apply the product rule to $2 \sqrt{6}$ .

$\frac{\sqrt{2^{2} {\sqrt{6}}^{2} - {(2 \sqrt{2})}^{2}}}{2 \sqrt{6}}$

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

$\frac{\sqrt{4 {\sqrt{6}}^{2} - {(2 \sqrt{2})}^{2}}}{2 \sqrt{6}}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{\sqrt{4 {(6^{\frac{1}{2}})}^{2} - {(2 \sqrt{2})}^{2}}}{2 \sqrt{6}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{4 \cdot 6^{\frac{1}{2} \cdot 2} - {(2 \sqrt{2})}^{2}}}{2 \sqrt{6}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{4 \cdot 6^{\frac{2}{2}} - {(2 \sqrt{2})}^{2}}}{2 \sqrt{6}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{4 \cdot 6^{\frac{2}{2}} - {(2 \sqrt{2})}^{2}}}{2 \sqrt{6}}$

Rewrite the expression.

$\frac{\sqrt{4 \cdot 6^{1} - {(2 \sqrt{2})}^{2}}}{2 \sqrt{6}}$

Evaluate the exponent.

$\frac{\sqrt{4 \cdot 6 - {(2 \sqrt{2})}^{2}}}{2 \sqrt{6}}$

Multiply $4$ by $6$ .

$\frac{\sqrt{24 - {(2 \sqrt{2})}^{2}}}{2 \sqrt{6}}$

Apply the product rule to $2 \sqrt{2}$ .

$\frac{\sqrt{24 - (2^{2} {\sqrt{2}}^{2})}}{2 \sqrt{6}}$

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

$\frac{\sqrt{24 - (4 {\sqrt{2}}^{2})}}{2 \sqrt{6}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{\sqrt{24 - (4 {(2^{\frac{1}{2}})}^{2})}}{2 \sqrt{6}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt{24 - (4 \cdot 2^{\frac{1}{2} \cdot 2})}}{2 \sqrt{6}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt{24 - (4 \cdot 2^{\frac{2}{2}})}}{2 \sqrt{6}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{24 - (4 \cdot 2^{\frac{2}{2}})}}{2 \sqrt{6}}$

Rewrite the expression.

$\frac{\sqrt{24 - (4 \cdot 2^{1})}}{2 \sqrt{6}}$

Evaluate the exponent.

$\frac{\sqrt{24 - (4 \cdot 2)}}{2 \sqrt{6}}$

Multiply $4$ by $2$ .

$\frac{\sqrt{24 - 1 \cdot 8}}{2 \sqrt{6}}$

Multiply $- 1$ by $8$ .

$\frac{\sqrt{24 - 8}}{2 \sqrt{6}}$

Subtract $8$ from $24$ .

$\frac{\sqrt{16}}{2 \sqrt{6}}$

Rewrite $16$ as $4^{2}$ .

$\frac{\sqrt{4^{2}}}{2 \sqrt{6}}$

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

$\frac{4}{2 \sqrt{6}}$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4$ .

$\frac{2 \cdot 2}{2 \sqrt{6}}$

Cancel the common factors.

Factor $2$ out of $2 \sqrt{6}$ .

$\frac{2 \cdot 2}{2 (\sqrt{6})}$

Cancel the common factor.

$\frac{2 \cdot 2}{2 \sqrt{6}}$

Rewrite the expression.

$\frac{2}{\sqrt{6}}$

Multiply $\frac{2}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{2}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{2 \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{2 \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{2 \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{2 \sqrt{6}}{{\sqrt{6}}^{2}}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{2 \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\frac{2 \sqrt{6}}{6^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 \sqrt{6}}{6^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{2 \sqrt{6}}{6^{1}}$

Evaluate the exponent.

$\frac{2 \sqrt{6}}{6}$

Cancel the common factor of $2$ and $6$ .

Factor $2$ out of $2 \sqrt{6}$ .

$\frac{2 (\sqrt{6})}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$\frac{2 \sqrt{6}}{2 \cdot 3}$

Cancel the common factor.

$\frac{2 \sqrt{6}}{2 \cdot 3}$

Rewrite the expression.

$\frac{\sqrt{6}}{3}$

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

Center: $(1, 4)$

${Vertex}_{1}$ : $(1, 4 + 2 \sqrt{6})$

${Vertex}_{2}$ : $(1, 4 - 2 \sqrt{6})$

${Focus}_{1}$ : $(1, 8)$

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

Eccentricity: $\frac{\sqrt{6}}{3}$

Find the Axis of Symmetry 3x^2+y^2-6x-8y-5=0

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