Find the Focus x=1/16(y-3)^2+9

$x = \frac{1}{16} {(y - 3)}^{2} + 9$

Step 1

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

$a = \frac{1}{16}$

$h = 9$

$k = 3$

Step 2

Find the vertex $(h, k)$ .

$(9, 3)$

Step 3

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.

$\frac{1}{4 \cdot \frac{1}{16}}$

Simplify.

Combine $4$ and $\frac{1}{16}$ .

$\frac{1}{\frac{4}{16}}$

Cancel the common factor of $4$ and $16$ .

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{16}}$

Cancel the common factors.

Factor $4$ out of $16$ .

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 4}}$

Cancel the common factor.

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 4}}$

Rewrite the expression.

$\frac{1}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 4$

Multiply $4$ by $1$ .

$4$

Step 4

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.

$(13, 3)$

Step 5