Find the Foci 9x^2+25y^2+100y-125=0

$9 x^{2} + 25 y^{2} + 100 y - 125 = 0$

Step 1

Find the standard form of the ellipse.

Add $125$ to both sides of the equation.

$9 x^{2} + 25 y^{2} + 100 y = 125$

Complete the square for $25 y^{2} + 100 y$ .

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 25$

$b = 100$

$c = 0$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{100}{2 \cdot 25}$

Simplify the right side.

Cancel the common factor of $100$ and $2$ .

Factor $2$ out of $100$ .

$d = \frac{2 \cdot 50}{2 \cdot 25}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 25$ .

$d = \frac{2 \cdot 50}{2 (25)}$

Cancel the common factor.

$d = \frac{2 \cdot 50}{2 \cdot 25}$

Rewrite the expression.

$d = \frac{50}{25}$

Cancel the common factor of $50$ and $25$ .

Factor $25$ out of $50$ .

$d = \frac{25 \cdot 2}{25}$

Cancel the common factors.

Factor $25$ out of $25$ .

$d = \frac{25 \cdot 2}{25 (1)}$

Cancel the common factor.

$d = \frac{25 \cdot 2}{25 \cdot 1}$

Rewrite the expression.

$d = \frac{2}{1}$

Divide $2$ by $1$ .

$d = 2$

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{100^{2}}{4 \cdot 25}$

Simplify the right side.

Simplify each term.

Raise $100$ to the power of $2$ .

$e = 0 - \frac{10000}{4 \cdot 25}$

Multiply $4$ by $25$ .

$e = 0 - \frac{10000}{100}$

Divide $10000$ by $100$ .

$e = 0 - 1 \cdot 100$

Multiply $- 1$ by $100$ .

$e = 0 - 100$

Subtract $100$ from $0$ .

$e = - 100$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $25 {(y + 2)}^{2} - 100$ .

$25 {(y + 2)}^{2} - 100$

Substitute $25 {(y + 2)}^{2} - 100$ for $25 y^{2} + 100 y$ in the equation $9 x^{2} + 25 y^{2} + 100 y = 125$ .

$9 x^{2} + 25 {(y + 2)}^{2} - 100 = 125$

Move $- 100$ to the right side of the equation by adding $100$ to both sides.

$9 x^{2} + 25 {(y + 2)}^{2} = 125 + 100$

Add $125$ and $100$ .

$9 x^{2} + 25 {(y + 2)}^{2} = 225$

Divide each term by $225$ to make the right side equal to one.

$\frac{9 x^{2}}{225} + \frac{25 {(y + 2)}^{2}}{225} = \frac{225}{225}$

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 $1$ .

$\frac{x^{2}}{25} + \frac{{(y + 2)}^{2}}{9} = 1$

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.

$a = 5$

$b = 3$

$k = - 2$

$h = 0$

Step 4

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.

$\sqrt{{(5)}^{2} - {(3)}^{2}}$

Simplify.

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

$\sqrt{25 - {(3)}^{2}}$

Raise $3$ to the power of $2$ .

$\sqrt{25 - 1 \cdot 9}$

Multiply $- 1$ by $9$ .

$\sqrt{25 - 9}$

Subtract $9$ from $25$ .

$\sqrt{16}$

Rewrite $16$ as $4^{2}$ .

$\sqrt{4^{2}}$

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

$4$

Step 5

Find the foci.

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

$(h + c, k)$

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

$(0 + 4, - 2)$

Simplify.

$(4, - 2)$

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

$(h - c, k)$

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

$(0 - (4), - 2)$

Simplify.

$(- 4, - 2)$

Ellipses have two foci.

${Focus}_{1}$ : $(4, - 2)$

${Focus}_{2}$ : $(- 4, - 2)$

${Focus}_{1}$ : $(4, - 2)$

${Focus}_{2}$ : $(- 4, - 2)$

Step 6

Find the Foci 9x^2+25y^2+100y-125=0

Related Questions

Support LCMGCF.com by disabling your adblocker.