Graph (2x^2)/25+((y-3)^2)/49=1

$\frac{2 x^{2}}{25} + \frac{{(y - 3)}^{2}}{49} = 1$

Step 1

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 $1$ .

$\frac{x^{2}}{\frac{25}{2}} + \frac{{(y - 3)}^{2}}{49} = 1$

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.

$a = 7$

$b = \frac{5 \sqrt{2}}{2}$

$k = 3$

$h = 0$

Step 4

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

$(0, 3)$

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.

$\sqrt{{(7)}^{2} - {(\frac{5 \sqrt{2}}{2})}^{2}}$

Simplify.

Raise $7$ to the power of $2$ .

$\sqrt{49 - {(\frac{5 \sqrt{2}}{2})}^{2}}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{5 \sqrt{2}}{2}$ .

$\sqrt{49 - \frac{{(5 \sqrt{2})}^{2}}{2^{2}}}$

Apply the product rule to $5 \sqrt{2}$ .

$\sqrt{49 - \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}}}$

Simplify the numerator.

Raise $5$ to the power of $2$ .

$\sqrt{49 - \frac{25 {\sqrt{2}}^{2}}{2^{2}}}$

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

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

$\sqrt{49 - \frac{25 {(2^{\frac{1}{2}})}^{2}}{2^{2}}}$

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

$\sqrt{49 - \frac{25 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}}}$

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

$\sqrt{49 - \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{49 - \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}}}$

Rewrite the expression.

$\sqrt{49 - \frac{25 \cdot 2^{1}}{2^{2}}}$

Evaluate the exponent.

$\sqrt{49 - \frac{25 \cdot 2}{2^{2}}}$

Reduce the expression by cancelling the common factors.

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

$\sqrt{49 - \frac{25 \cdot 2}{4}}$

Multiply $25$ by $2$ .

$\sqrt{49 - \frac{50}{4}}$

Cancel the common factor of $50$ and $4$ .

Factor $2$ out of $50$ .

$\sqrt{49 - \frac{2 (25)}{4}}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\sqrt{49 - \frac{2 \cdot 25}{2 \cdot 2}}$

Cancel the common factor.

$\sqrt{49 - \frac{2 \cdot 25}{2 \cdot 2}}$

Rewrite the expression.

$\sqrt{49 - \frac{25}{2}}$

To write $49$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sqrt{49 \cdot \frac{2}{2} - \frac{25}{2}}$

Combine $49$ and $\frac{2}{2}$ .

$\sqrt{\frac{49 \cdot 2}{2} - \frac{25}{2}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{49 \cdot 2 - 25}{2}}$

Simplify the numerator.

Multiply $49$ by $2$ .

$\sqrt{\frac{98 - 25}{2}}$

Subtract $25$ from $98$ .

$\sqrt{\frac{73}{2}}$

Rewrite $\sqrt{\frac{73}{2}}$ as $\frac{\sqrt{73}}{\sqrt{2}}$ .

$\frac{\sqrt{73}}{\sqrt{2}}$

Multiply $\frac{\sqrt{73}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{73}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{73}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\sqrt{73} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

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

$\frac{\sqrt{73} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

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

$\frac{\sqrt{73} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

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

$\frac{\sqrt{73} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{\sqrt{73} \sqrt{2}}{{\sqrt{2}}^{2}}$

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

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

$\frac{\sqrt{73} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

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

$\frac{\sqrt{73} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

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

$\frac{\sqrt{73} \sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{73} \sqrt{2}}{2^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{\sqrt{73} \sqrt{2}}{2^{1}}$

Evaluate the exponent.

$\frac{\sqrt{73} \sqrt{2}}{2}$

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{\sqrt{73 \cdot 2}}{2}$

Multiply $73$ by $2$ .

$\frac{\sqrt{146}}{2}$

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.

$(0, 3 + 7)$

Simplify.

$(0, 10)$

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.

$(0, 3 - (7))$

Simplify.

$(0, - 4)$

Ellipses have two vertices.

${Vertex}_{1}$ : $(0, 10)$

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

${Vertex}_{1}$ : $(0, 10)$

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

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.

$(0, 3 + \frac{\sqrt{146}}{2})$

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.

$(0, 3 - (\frac{\sqrt{146}}{2}))$

Simplify.

$(0, 3 - \frac{\sqrt{146}}{2})$

Ellipses have two foci.

${Focus}_{1}$ : $(0, 3 + \frac{\sqrt{146}}{2})$

${Focus}_{2}$ : $(0, 3 - \frac{\sqrt{146}}{2})$

${Focus}_{1}$ : $(0, 3 + \frac{\sqrt{146}}{2})$

${Focus}_{2}$ : $(0, 3 - \frac{\sqrt{146}}{2})$

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.

$\frac{\sqrt{{(7)}^{2} - {(\frac{5 \sqrt{2}}{2})}^{2}}}{7}$

Simplify.

Simplify the numerator.

Raise $7$ to the power of $2$ .

$\frac{\sqrt{49 - {(\frac{5 \sqrt{2}}{2})}^{2}}}{7}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $\frac{5 \sqrt{2}}{2}$ .

$\frac{\sqrt{49 - \frac{{(5 \sqrt{2})}^{2}}{2^{2}}}}{7}$

Apply the product rule to $5 \sqrt{2}$ .

$\frac{\sqrt{49 - \frac{5^{2} {\sqrt{2}}^{2}}{2^{2}}}}{7}$

Simplify the numerator.

Raise $5$ to the power of $2$ .

$\frac{\sqrt{49 - \frac{25 {\sqrt{2}}^{2}}{2^{2}}}}{7}$

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

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

$\frac{\sqrt{49 - \frac{25 {(2^{\frac{1}{2}})}^{2}}{2^{2}}}}{7}$

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

$\frac{\sqrt{49 - \frac{25 \cdot 2^{\frac{1}{2} \cdot 2}}{2^{2}}}}{7}$

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

$\frac{\sqrt{49 - \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}}}}{7}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{49 - \frac{25 \cdot 2^{\frac{2}{2}}}{2^{2}}}}{7}$

Rewrite the expression.

$\frac{\sqrt{49 - \frac{25 \cdot 2^{1}}{2^{2}}}}{7}$

Evaluate the exponent.

$\frac{\sqrt{49 - \frac{25 \cdot 2}{2^{2}}}}{7}$

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

$\frac{\sqrt{49 - \frac{25 \cdot 2}{4}}}{7}$

Multiply $25$ by $2$ .

$\frac{\sqrt{49 - \frac{50}{4}}}{7}$

Cancel the common factor of $50$ and $4$ .

Factor $2$ out of $50$ .

$\frac{\sqrt{49 - \frac{2 (25)}{4}}}{7}$

Cancel the common factors.

Factor $2$ out of $4$ .

$\frac{\sqrt{49 - \frac{2 \cdot 25}{2 \cdot 2}}}{7}$

Cancel the common factor.

$\frac{\sqrt{49 - \frac{2 \cdot 25}{2 \cdot 2}}}{7}$

Rewrite the expression.

$\frac{\sqrt{49 - \frac{25}{2}}}{7}$

To write $49$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\sqrt{49 \cdot \frac{2}{2} - \frac{25}{2}}}{7}$

Combine $49$ and $\frac{2}{2}$ .

$\frac{\sqrt{\frac{49 \cdot 2}{2} - \frac{25}{2}}}{7}$

Combine the numerators over the common denominator.

$\frac{\sqrt{\frac{49 \cdot 2 - 25}{2}}}{7}$

Simplify the numerator.

Multiply $49$ by $2$ .

$\frac{\sqrt{\frac{98 - 25}{2}}}{7}$

Subtract $25$ from $98$ .

$\frac{\sqrt{\frac{73}{2}}}{7}$

Rewrite $\sqrt{\frac{73}{2}}$ as $\frac{\sqrt{73}}{\sqrt{2}}$ .

$\frac{\frac{\sqrt{73}}{\sqrt{2}}}{7}$

Multiply $\frac{\sqrt{73}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\frac{\sqrt{73}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}}{7}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{73}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\frac{\sqrt{73} \sqrt{2}}{\sqrt{2} \sqrt{2}}}{7}$

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

$\frac{\frac{\sqrt{73} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}}{7}$

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

$\frac{\frac{\sqrt{73} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}}{7}$

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

$\frac{\frac{\sqrt{73} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}}{7}$

Add $1$ and $1$ .

$\frac{\frac{\sqrt{73} \sqrt{2}}{{\sqrt{2}}^{2}}}{7}$

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

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

$\frac{\frac{\sqrt{73} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}}{7}$

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

$\frac{\frac{\sqrt{73} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}}{7}$

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

$\frac{\frac{\sqrt{73} \sqrt{2}}{2^{\frac{2}{2}}}}{7}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\frac{\sqrt{73} \sqrt{2}}{2^{\frac{2}{2}}}}{7}$

Rewrite the expression.

$\frac{\frac{\sqrt{73} \sqrt{2}}{2^{1}}}{7}$

Evaluate the exponent.

$\frac{\frac{\sqrt{73} \sqrt{2}}{2}}{7}$

Simplify the numerator.

Combine using the product rule for radicals.

$\frac{\frac{\sqrt{73 \cdot 2}}{2}}{7}$

Multiply $73$ by $2$ .

$\frac{\frac{\sqrt{146}}{2}}{7}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{146}}{2} \cdot \frac{1}{7}$

Multiply $\frac{\sqrt{146}}{2} \cdot \frac{1}{7}$ .

Multiply $\frac{\sqrt{146}}{2}$ by $\frac{1}{7}$ .

$\frac{\sqrt{146}}{2 \cdot 7}$

Multiply $2$ by $7$ .

$\frac{\sqrt{146}}{14}$

Step 9

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

Center: $(0, 3)$

${Vertex}_{1}$ : $(0, 10)$

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

${Focus}_{1}$ : $(0, 3 + \frac{\sqrt{146}}{2})$

${Focus}_{2}$ : $(0, 3 - \frac{\sqrt{146}}{2})$

Eccentricity: $\frac{\sqrt{146}}{14}$

Step 10

Graph (2x^2)/25+((y-3)^2)/49=1

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