Find the Properties 9x^2+7y^2=42

$9 x^{2} + 7 y^{2} = 42$

Step 1

Find the standard form of the ellipse.

Divide each term by $42$ to make the right side equal to one.

$\frac{9 x^{2}}{42} + \frac{7 y^{2}}{42} = \frac{42}{42}$

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 $1$ .

$\frac{x^{2}}{\frac{14}{3}} + \frac{y^{2}}{6} = 1$

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.

$a = \sqrt{6}$

$b = \frac{\sqrt{42}}{3}$

$k = 0$

$h = 0$

Step 4

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

$(0, 0)$

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.

$\sqrt{{(\sqrt{6})}^{2} - {(\frac{\sqrt{42}}{3})}^{2}}$

Simplify.

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

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

$\sqrt{{(6^{\frac{1}{2}})}^{2} - {(\frac{\sqrt{42}}{3})}^{2}}$

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

$\sqrt{6^{\frac{1}{2} \cdot 2} - {(\frac{\sqrt{42}}{3})}^{2}}$

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

$\sqrt{6^{\frac{2}{2}} - {(\frac{\sqrt{42}}{3})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{6^{\frac{2}{2}} - {(\frac{\sqrt{42}}{3})}^{2}}$

Rewrite the expression.

$\sqrt{6^{1} - {(\frac{\sqrt{42}}{3})}^{2}}$

Evaluate the exponent.

$\sqrt{6 - {(\frac{\sqrt{42}}{3})}^{2}}$

Apply the product rule to $\frac{\sqrt{42}}{3}$ .

$\sqrt{6 - \frac{{\sqrt{42}}^{2}}{3^{2}}}$

Rewrite ${\sqrt{42}}^{2}$ as $42$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{42}$ as $42^{\frac{1}{2}}$ .

$\sqrt{6 - \frac{{(42^{\frac{1}{2}})}^{2}}{3^{2}}}$

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

$\sqrt{6 - \frac{42^{\frac{1}{2} \cdot 2}}{3^{2}}}$

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

$\sqrt{6 - \frac{42^{\frac{2}{2}}}{3^{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{6 - \frac{42^{\frac{2}{2}}}{3^{2}}}$

Rewrite the expression.

$\sqrt{6 - \frac{42^{1}}{3^{2}}}$

Evaluate the exponent.

$\sqrt{6 - \frac{42}{3^{2}}}$

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

$\sqrt{6 - \frac{42}{9}}$

Cancel the common factor of $42$ and $9$ .

Factor $3$ out of $42$ .

$\sqrt{6 - \frac{3 (14)}{9}}$

Cancel the common factors.

Factor $3$ out of $9$ .

$\sqrt{6 - \frac{3 \cdot 14}{3 \cdot 3}}$

Cancel the common factor.

$\sqrt{6 - \frac{3 \cdot 14}{3 \cdot 3}}$

Rewrite the expression.

$\sqrt{6 - \frac{14}{3}}$

To write $6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\sqrt{6 \cdot \frac{3}{3} - \frac{14}{3}}$

Combine $6$ and $\frac{3}{3}$ .

$\sqrt{\frac{6 \cdot 3}{3} - \frac{14}{3}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{6 \cdot 3 - 14}{3}}$

Simplify the numerator.

Multiply $6$ by $3$ .

$\sqrt{\frac{18 - 14}{3}}$

Subtract $14$ from $18$ .

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

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

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

Simplify the numerator.

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

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

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

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

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

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

$\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 \sqrt{3}}{3^{1}}$

Evaluate the exponent.

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

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.

$(0, 0 + \sqrt{6})$

Simplify.

$(0, \sqrt{6})$

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.

$(0, 0 - (\sqrt{6}))$

Simplify.

$(0, - \sqrt{6})$

Ellipses have two vertices.

${Vertex}_{1}$ : $(0, \sqrt{6})$

${Vertex}_{2}$ : $(0, - \sqrt{6})$

${Vertex}_{1}$ : $(0, \sqrt{6})$

${Vertex}_{2}$ : $(0, - \sqrt{6})$

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.

$(0, 0 + \frac{2 \sqrt{3}}{3})$

Simplify.

$(0, \frac{2 \sqrt{3}}{3})$

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.

$(0, 0 - (\frac{2 \sqrt{3}}{3}))$

Simplify.

$(0, - \frac{2 \sqrt{3}}{3})$

Ellipses have two foci.

${Focus}_{1}$ : $(0, \frac{2 \sqrt{3}}{3})$

${Focus}_{2}$ : $(0, - \frac{2 \sqrt{3}}{3})$

${Focus}_{1}$ : $(0, \frac{2 \sqrt{3}}{3})$

${Focus}_{2}$ : $(0, - \frac{2 \sqrt{3}}{3})$

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.

$\frac{\sqrt{{(\sqrt{6})}^{2} - {(\frac{\sqrt{42}}{3})}^{2}}}{\sqrt{6}}$

Simplify.

Simplify the numerator.

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

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

$\frac{\sqrt{{(6^{\frac{1}{2}})}^{2} - {(\frac{\sqrt{42}}{3})}^{2}}}{\sqrt{6}}$

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

$\frac{\sqrt{6^{\frac{1}{2} \cdot 2} - {(\frac{\sqrt{42}}{3})}^{2}}}{\sqrt{6}}$

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

$\frac{\sqrt{6^{\frac{2}{2}} - {(\frac{\sqrt{42}}{3})}^{2}}}{\sqrt{6}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{6^{\frac{2}{2}} - {(\frac{\sqrt{42}}{3})}^{2}}}{\sqrt{6}}$

Rewrite the expression.

$\frac{\sqrt{6^{1} - {(\frac{\sqrt{42}}{3})}^{2}}}{\sqrt{6}}$

Evaluate the exponent.

$\frac{\sqrt{6 - {(\frac{\sqrt{42}}{3})}^{2}}}{\sqrt{6}}$

Apply the product rule to $\frac{\sqrt{42}}{3}$ .

$\frac{\sqrt{6 - \frac{{\sqrt{42}}^{2}}{3^{2}}}}{\sqrt{6}}$

Rewrite ${\sqrt{42}}^{2}$ as $42$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{42}$ as $42^{\frac{1}{2}}$ .

$\frac{\sqrt{6 - \frac{{(42^{\frac{1}{2}})}^{2}}{3^{2}}}}{\sqrt{6}}$

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

$\frac{\sqrt{6 - \frac{42^{\frac{1}{2} \cdot 2}}{3^{2}}}}{\sqrt{6}}$

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

$\frac{\sqrt{6 - \frac{42^{\frac{2}{2}}}{3^{2}}}}{\sqrt{6}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{6 - \frac{42^{\frac{2}{2}}}{3^{2}}}}{\sqrt{6}}$

Rewrite the expression.

$\frac{\sqrt{6 - \frac{42^{1}}{3^{2}}}}{\sqrt{6}}$

Evaluate the exponent.

$\frac{\sqrt{6 - \frac{42}{3^{2}}}}{\sqrt{6}}$

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

$\frac{\sqrt{6 - \frac{42}{9}}}{\sqrt{6}}$

Cancel the common factor of $42$ and $9$ .

Factor $3$ out of $42$ .

$\frac{\sqrt{6 - \frac{3 (14)}{9}}}{\sqrt{6}}$

Cancel the common factors.

Factor $3$ out of $9$ .

$\frac{\sqrt{6 - \frac{3 \cdot 14}{3 \cdot 3}}}{\sqrt{6}}$

Cancel the common factor.

$\frac{\sqrt{6 - \frac{3 \cdot 14}{3 \cdot 3}}}{\sqrt{6}}$

Rewrite the expression.

$\frac{\sqrt{6 - \frac{14}{3}}}{\sqrt{6}}$

To write $6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\sqrt{6 \cdot \frac{3}{3} - \frac{14}{3}}}{\sqrt{6}}$

Combine $6$ and $\frac{3}{3}$ .

$\frac{\sqrt{\frac{6 \cdot 3}{3} - \frac{14}{3}}}{\sqrt{6}}$

Combine the numerators over the common denominator.

$\frac{\sqrt{\frac{6 \cdot 3 - 14}{3}}}{\sqrt{6}}$

Simplify the numerator.

Multiply $6$ by $3$ .

$\frac{\sqrt{\frac{18 - 14}{3}}}{\sqrt{6}}$

Subtract $14$ from $18$ .

$\frac{\sqrt{\frac{4}{3}}}{\sqrt{6}}$

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

$\frac{\frac{\sqrt{4}}{\sqrt{3}}}{\sqrt{6}}$

Simplify the numerator.

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

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

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

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

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}}{\sqrt{6}}$

Raise $\sqrt{3}$ to the power of $1$ .

$\frac{\frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}}{\sqrt{6}}$

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

$\frac{\frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}}{\sqrt{6}}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{\frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}}{\sqrt{6}}$

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

$\frac{\frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}}{\sqrt{6}}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{\frac{2 \sqrt{3}}{3^{1}}}{\sqrt{6}}$

Evaluate the exponent.

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

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 \sqrt{3}}{3} \cdot \frac{1}{\sqrt{6}}$

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\frac{2 \sqrt{3}}{3} (\frac{1}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})$

Simplify terms.

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

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

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

$\frac{2 \sqrt{3}}{3} \cdot \frac{\sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

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

$\frac{2 \sqrt{3}}{3} \cdot \frac{\sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

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

$\frac{2 \sqrt{3}}{3} \cdot \frac{\sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Add $1$ and $1$ .

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

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

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

$\frac{2 \sqrt{3}}{3} \cdot \frac{\sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

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

$\frac{2 \sqrt{3}}{3} \cdot \frac{\sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{2 \sqrt{3}}{3} \cdot \frac{\sqrt{6}}{6^{1}}$

Evaluate the exponent.

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

Cancel the common factor of $2$ .

Factor $2$ out of $2 \sqrt{3}$ .

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

Factor $2$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{\sqrt{3}}{3}$ by $\frac{\sqrt{6}}{3}$ .

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

Combine using the product rule for radicals.

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

Multiply.

Multiply $3$ by $6$ .

$\frac{\sqrt{18}}{3 \cdot 3}$

Multiply $3$ by $3$ .

$\frac{\sqrt{18}}{9}$

Simplify the numerator.

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$\frac{\sqrt{9 (2)}}{9}$

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

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

Pull terms out from under the radical.

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

Cancel the common factor of $3$ and $9$ .

Factor $3$ out of $3 \sqrt{2}$ .

$\frac{3 (\sqrt{2})}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 9

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

Center: $(0, 0)$

${Vertex}_{1}$ : $(0, \sqrt{6})$

${Vertex}_{2}$ : $(0, - \sqrt{6})$

${Focus}_{1}$ : $(0, \frac{2 \sqrt{3}}{3})$

${Focus}_{2}$ : $(0, - \frac{2 \sqrt{3}}{3})$

Eccentricity: $\frac{\sqrt{2}}{3}$

Step 10

Find the Properties 9x^2+7y^2=42

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