Find the Foci 8x^2+8y^2-56x+98=0

$8 x^{2} + 8 y^{2} - 56 x + 98 = 0$

Step 1

Subtract $98$ from both sides of the equation.

$8 x^{2} + 8 y^{2} - 56 x = - 98$

Step 2

Complete the square for $8 x^{2} - 56 x$ .

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

$a = 8$

$b = - 56$

$c = 0$

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 $d = \frac{b}{2 a}$ .

$d = \frac{- 56}{2 \cdot 8}$

Simplify the right side.

Cancel the common factor of $- 56$ and $2$ .

Factor $2$ out of $- 56$ .

$d = \frac{2 \cdot - 28}{2 \cdot 8}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 8$ .

$d = \frac{2 \cdot - 28}{2 (8)}$

Cancel the common factor.

$d = \frac{2 \cdot - 28}{2 \cdot 8}$

Rewrite the expression.

$d = \frac{- 28}{8}$

Cancel the common factor of $- 28$ and $8$ .

Factor $4$ out of $- 28$ .

$d = \frac{4 (- 7)}{8}$

Cancel the common factors.

Factor $4$ out of $8$ .

$d = \frac{4 \cdot - 7}{4 \cdot 2}$

Cancel the common factor.

$d = \frac{4 \cdot - 7}{4 \cdot 2}$

Rewrite the expression.

$d = \frac{- 7}{2}$

Move the negative in front of the fraction.

$d = - \frac{7}{2}$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 56)}^{2}}{4 \cdot 8}$

Simplify the right side.

Simplify each term.

Raise $- 56$ to the power of $2$ .

$e = 0 - \frac{3136}{4 \cdot 8}$

Multiply $4$ by $8$ .

$e = 0 - \frac{3136}{32}$

Divide $3136$ by $32$ .

$e = 0 - 1 \cdot 98$

Multiply $- 1$ by $98$ .

$e = 0 - 98$

Subtract $98$ from $0$ .

$e = - 98$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $8 {(x - \frac{7}{2})}^{2} - 98$ .

$8 {(x - \frac{7}{2})}^{2} - 98$

Step 3

Substitute $8 {(x - \frac{7}{2})}^{2} - 98$ for $8 x^{2} - 56 x$ in the equation $8 x^{2} + 8 y^{2} - 56 x = - 98$ .

$8 {(x - \frac{7}{2})}^{2} - 98 + 8 y^{2} = - 98$

Step 4

Move $- 98$ to the right side of the equation by adding $98$ to both sides.

$8 {(x - \frac{7}{2})}^{2} + 8 y^{2} = - 98 + 98$

Step 5

Add $- 98$ and $98$ .

$8 {(x - \frac{7}{2})}^{2} + 8 y^{2} = 0$

Step 6

Divide each term by $0$ to make the right side equal to one.

$\frac{8 {(x - \frac{7}{2})}^{2}}{0} + \frac{8 y^{2}}{0} = \frac{0}{0}$

Step 7

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 $1$ .

${(x - \frac{7}{2})}^{2} + y^{2} = 1$

Find the Foci 8x^2+8y^2-56x+98=0

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