Find the Focus (y+5/2)^2=-5(x-9/2)

${(y + \frac{5}{2})}^{2} = - 5 (x - \frac{9}{2})$

Step 1

Isolate $x$ to the left side of the equation.

Rewrite the equation as $- 5 (x - \frac{9}{2}) = {(y + \frac{5}{2})}^{2}$ .

$- 5 (x - \frac{9}{2}) = {(y + \frac{5}{2})}^{2}$

Divide each term in $- 5 (x - \frac{9}{2}) = {(y + \frac{5}{2})}^{2}$ by $- 5$ and simplify.

Divide each term in $- 5 (x - \frac{9}{2}) = {(y + \frac{5}{2})}^{2}$ by $- 5$ .

$\frac{- 5 (x - \frac{9}{2})}{- 5} = \frac{{(y + \frac{5}{2})}^{2}}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 (x - \frac{9}{2})}{- 5} = \frac{{(y + \frac{5}{2})}^{2}}{- 5}$

Divide $x - \frac{9}{2}$ by $1$ .

$x - \frac{9}{2} = \frac{{(y + \frac{5}{2})}^{2}}{- 5}$

Simplify the right side.

Move the negative in front of the fraction.

$x - \frac{9}{2} = - \frac{{(y + \frac{5}{2})}^{2}}{5}$

Add $\frac{9}{2}$ to both sides of the equation.

$x = - \frac{{(y + \frac{5}{2})}^{2}}{5} + \frac{9}{2}$

Reorder terms.

$x = - \frac{1}{5} \cdot {(y + \frac{5}{2})}^{2} + \frac{9}{2}$

Step 2

Use the vertex form, $x = a {(y - k)}^{2} + h$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - \frac{1}{5}$

$h = \frac{9}{2}$

$k = - \frac{5}{2}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{9}{2}, - \frac{5}{2})$

Step 4

Find $p$ , the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot (- \frac{1}{5})}$

Simplify.

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{4 \cdot (- \frac{1}{5})}$

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{1}{5})}$

Combine $4$ and $\frac{1}{5}$ .

$- \frac{1}{\frac{4}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{5}{4}))$

Multiply $\frac{5}{4}$ by $1$ .

$- \frac{5}{4}$

Step 5

Find the focus.

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

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

$(\frac{13}{4}, - \frac{5}{2})$

Step 6

Find the Focus (y+5/2)^2=-5(x-9/2)

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