Find the Eccentricity 24x^2+25y^2=600

$24 x^{2} + 25 y^{2} = 600$

Step 1

Divide each term by $600$ to make the right side equal to one.

$\frac{24 x^{2}}{600} + \frac{25 y^{2}}{600} = \frac{600}{600}$

Step 2

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 $1$ .

$\frac{x^{2}}{25} + \frac{y^{2}}{24} = 1$

Step 3

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.

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

Step 4

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.

$a = 5$

$b = 2 \sqrt{6}$

$k = 0$

$h = 0$

Step 5

Find the eccentricity by using the following formula.

$\frac{\sqrt{a^{2} - b^{2}}}{a}$

Step 6

Substitute the values of $a$ and $b$ into the formula.

$\frac{\sqrt{{(5)}^{2} - {(2 \sqrt{6})}^{2}}}{5}$

Step 7

Simplify the numerator.

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

$\frac{\sqrt{25 - {(2 \sqrt{6})}^{2}}}{5}$

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

$\frac{\sqrt{25 - (2^{2} {\sqrt{6}}^{2})}}{5}$

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

$\frac{\sqrt{25 - (4 {\sqrt{6}}^{2})}}{5}$

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

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

$\frac{\sqrt{25 - (4 {(6^{\frac{1}{2}})}^{2})}}{5}$

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

$\frac{\sqrt{25 - (4 \cdot 6^{\frac{1}{2} \cdot 2})}}{5}$

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

$\frac{\sqrt{25 - (4 \cdot 6^{\frac{2}{2}})}}{5}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{\sqrt{25 - (4 \cdot 6^{\frac{2}{2}})}}{5}$

Rewrite the expression.

$\frac{\sqrt{25 - (4 \cdot 6^{1})}}{5}$

Evaluate the exponent.

$\frac{\sqrt{25 - (4 \cdot 6)}}{5}$

Multiply $- (4 \cdot 6)$ .

Multiply $4$ by $6$ .

$\frac{\sqrt{25 - 1 \cdot 24}}{5}$

Multiply $- 1$ by $24$ .

$\frac{\sqrt{25 - 24}}{5}$

Subtract $24$ from $25$ .

$\frac{\sqrt{1}}{5}$

Any root of $1$ is $1$ .

$\frac{1}{5}$

Step 8

Find the Eccentricity 24x^2+25y^2=600

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