Find the Properties 8x^2=224-7y^2

$8 x^{2} = 224 - 7 y^{2}$

Find the standard form of the ellipse.

Add $7 y^{2}$ to both sides of the equation.

$8 x^{2} + 7 y^{2} = 224$

Divide each term by $224$ to make the right side equal to one.

$\frac{8 x^{2}}{224} + \frac{7 y^{2}}{224} = \frac{224}{224}$

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 $1$ .

$\frac{x^{2}}{28} + \frac{y^{2}}{32} = 1$

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.

$a = 4 \sqrt{2}$

$b = 2 \sqrt{7}$

$k = 0$

$h = 0$

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

$(0, 0)$

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.

$\sqrt{{(4 \sqrt{2})}^{2} - {(2 \sqrt{7})}^{2}}$

Simplify.

Simplify the expression.

Apply the product rule to $4 \sqrt{2}$ .

$\sqrt{4^{2} {\sqrt{2}}^{2} - {(2 \sqrt{7})}^{2}}$

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

$\sqrt{16 {\sqrt{2}}^{2} - {(2 \sqrt{7})}^{2}}$

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

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

$\sqrt{16 {(2^{\frac{1}{2}})}^{2} - {(2 \sqrt{7})}^{2}}$

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

$\sqrt{16 \cdot 2^{\frac{1}{2} \cdot 2} - {(2 \sqrt{7})}^{2}}$

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

$\sqrt{16 \cdot 2^{\frac{2}{2}} - {(2 \sqrt{7})}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{16 \cdot 2^{\frac{2}{2}} - {(2 \sqrt{7})}^{2}}$

Rewrite the expression.

$\sqrt{16 \cdot 2^{1} - {(2 \sqrt{7})}^{2}}$

Evaluate the exponent.

$\sqrt{16 \cdot 2 - {(2 \sqrt{7})}^{2}}$

Simplify the expression.

Multiply $16$ by $2$ .

$\sqrt{32 - {(2 \sqrt{7})}^{2}}$

Apply the product rule to $2 \sqrt{7}$ .

$\sqrt{32 - (2^{2} {\sqrt{7}}^{2})}$

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

$\sqrt{32 - (4 {\sqrt{7}}^{2})}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sqrt{32 - (4 {(7^{\frac{1}{2}})}^{2})}$

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

$\sqrt{32 - (4 \cdot 7^{\frac{1}{2} \cdot 2})}$

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

$\sqrt{32 - (4 \cdot 7^{\frac{2}{2}})}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{32 - (4 \cdot 7^{\frac{2}{2}})}$

Rewrite the expression.

$\sqrt{32 - (4 \cdot 7^{1})}$

Evaluate the exponent.

$\sqrt{32 - (4 \cdot 7)}$

Simplify the expression.

Multiply $4$ by $7$ .

$\sqrt{32 - 1 \cdot 28}$

Multiply $- 1$ by $28$ .

$\sqrt{32 - 28}$

Subtract $28$ from $32$ .

$\sqrt{4}$

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

$\sqrt{2^{2}}$

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

$2$

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.

$(0, 0 + 4 \sqrt{2})$

Simplify.

$(0, 4 \sqrt{2})$

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.

$(0, 0 - (4 \sqrt{2}))$

Simplify.

$(0, - 4 \sqrt{2})$

Ellipses have two vertices.

${Vertex}_{1}$ : $(0, 4 \sqrt{2})$

${Vertex}_{2}$ : $(0, - 4 \sqrt{2})$

${Vertex}_{1}$ : $(0, 4 \sqrt{2})$

${Vertex}_{2}$ : $(0, - 4 \sqrt{2})$

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.

$(0, 0 + 2)$

Simplify.

$(0, 2)$

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.

$(0, 0 - (2))$

Simplify.

$(0, - 2)$

Ellipses have two foci.

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

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

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

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

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.

$\frac{\sqrt{{(4 \sqrt{2})}^{2} - {(2 \sqrt{7})}^{2}}}{4 \sqrt{2}}$

Simplify.

Simplify the numerator.

Apply the product rule to $4 \sqrt{2}$ .

$\frac{\sqrt{4^{2} {\sqrt{2}}^{2} - {(2 \sqrt{7})}^{2}}}{4 \sqrt{2}}$

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

$\frac{\sqrt{16 {\sqrt{2}}^{2} - {(2 \sqrt{7})}^{2}}}{4 \sqrt{2}}$

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

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

$\frac{\sqrt{16 {(2^{\frac{1}{2}})}^{2} - {(2 \sqrt{7})}^{2}}}{4 \sqrt{2}}$

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

$\frac{\sqrt{16 \cdot 2^{\frac{1}{2} \cdot 2} - {(2 \sqrt{7})}^{2}}}{4 \sqrt{2}}$

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

$\frac{\sqrt{16 \cdot 2^{\frac{2}{2}} - {(2 \sqrt{7})}^{2}}}{4 \sqrt{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{16 \cdot 2^{\frac{2}{2}} - {(2 \sqrt{7})}^{2}}}{4 \sqrt{2}}$

Rewrite the expression.

$\frac{\sqrt{16 \cdot 2^{1} - {(2 \sqrt{7})}^{2}}}{4 \sqrt{2}}$

Evaluate the exponent.

$\frac{\sqrt{16 \cdot 2 - {(2 \sqrt{7})}^{2}}}{4 \sqrt{2}}$

Multiply $16$ by $2$ .

$\frac{\sqrt{32 - {(2 \sqrt{7})}^{2}}}{4 \sqrt{2}}$

Apply the product rule to $2 \sqrt{7}$ .

$\frac{\sqrt{32 - (2^{2} {\sqrt{7}}^{2})}}{4 \sqrt{2}}$

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

$\frac{\sqrt{32 - (4 {\sqrt{7}}^{2})}}{4 \sqrt{2}}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{\sqrt{32 - (4 {(7^{\frac{1}{2}})}^{2})}}{4 \sqrt{2}}$

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

$\frac{\sqrt{32 - (4 \cdot 7^{\frac{1}{2} \cdot 2})}}{4 \sqrt{2}}$

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

$\frac{\sqrt{32 - (4 \cdot 7^{\frac{2}{2}})}}{4 \sqrt{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{32 - (4 \cdot 7^{\frac{2}{2}})}}{4 \sqrt{2}}$

Rewrite the expression.

$\frac{\sqrt{32 - (4 \cdot 7^{1})}}{4 \sqrt{2}}$

Evaluate the exponent.

$\frac{\sqrt{32 - (4 \cdot 7)}}{4 \sqrt{2}}$

Multiply $4$ by $7$ .

$\frac{\sqrt{32 - 1 \cdot 28}}{4 \sqrt{2}}$

Multiply $- 1$ by $28$ .

$\frac{\sqrt{32 - 28}}{4 \sqrt{2}}$

Subtract $28$ from $32$ .

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

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

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

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

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

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

Factor $2$ out of $2$ .

$\frac{2 (1)}{4 \sqrt{2}}$

Cancel the common factors.

Factor $2$ out of $4 \sqrt{2}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $\frac{1}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{1}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Move $\sqrt{2}$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $2$ by $2$ .

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

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

Center: $(0, 0)$

${Vertex}_{1}$ : $(0, 4 \sqrt{2})$

${Vertex}_{2}$ : $(0, - 4 \sqrt{2})$

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

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

Eccentricity: $\frac{\sqrt{2}}{4}$

Find the Properties 8x^2=224-7y^2

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