Precalculus Examples

$\frac{x^{2}}{100} + \frac{y^{2}}{64} = 1$

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 $1$ .

$\frac{x^{2}}{100} + \frac{y^{2}}{64} = 1$

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.

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

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.

$a = 10$

$b = 8$

$k = 0$

$h = 0$

Step 4

Find $c$ , the distance from the center to a focus.

Tap for more steps...

Step 4.1

Find the distance from the center to a focus of the ellipse by using the following formula.

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

Step 4.2

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

$\sqrt{{(10)}^{2} - {(8)}^{2}}$

Step 4.3

Simplify.

Tap for more steps...

Step 4.3.1

Raise $10$ to the power of $2$ .

$\sqrt{100 - {(8)}^{2}}$

Step 4.3.2

Raise $8$ to the power of $2$ .

$\sqrt{100 - 1 \cdot 64}$

Step 4.3.3

Multiply $- 1$ by $64$ .

$\sqrt{100 - 64}$

Step 4.3.4

Subtract $64$ from $100$ .

$\sqrt{36}$

Step 4.3.5

Rewrite $36$ as $6^{2}$ .

$\sqrt{6^{2}}$

Step 4.3.6

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

$6$

Step 5

Find the foci.

Tap for more steps...

Step 5.1

The first focus of an ellipse can be found by adding $c$ to $h$ .

$(h + c, k)$

Step 5.2

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(0 + 6, 0)$

Step 5.3

Simplify.

$(6, 0)$

Step 5.4

The second focus of an ellipse can be found by subtracting $c$ from $h$ .

$(h - c, k)$

Step 5.5

Substitute the known values of $h$ , $c$ , and $k$ into the formula.

$(0 - (6), 0)$

Step 5.6

Simplify.

$(- 6, 0)$

Step 5.7

Ellipses have two foci.

${Focus}_{1}$ : $(6, 0)$

${Focus}_{2}$ : $(- 6, 0)$

${Focus}_{1}$ : $(6, 0)$

${Focus}_{2}$ : $(- 6, 0)$

Step 6