Find the Eccentricity 7x^2+16y^2=112

$7 x^{2} + 16 y^{2} = 112$

Step 1

Divide each term by $112$ to make the right side equal to one.

$\frac{7 x^{2}}{112} + \frac{16 y^{2}}{112} = \frac{112}{112}$

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 $1$ .

$\frac{x^{2}}{16} + \frac{y^{2}}{7} = 1$

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.

$a = 4$

$b = \sqrt{7}$

$k = 0$

$h = 0$

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.

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

Step 7

Simplify the numerator.

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$\frac{\sqrt{16 - 1 \cdot 7}}{4}$

Multiply $- 1$ by $7$ .

$\frac{\sqrt{16 - 7}}{4}$

Subtract $7$ from $16$ .

$\frac{\sqrt{9}}{4}$

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

$\frac{\sqrt{3^{2}}}{4}$

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

$\frac{3}{4}$

Step 8

Find the Eccentricity 7x^2+16y^2=112

Related Questions

Support LCMGCF.com by disabling your adblocker.