Find the Focus x-7=3/4y^2

$x - 7 = \frac{3}{4} y^{2}$

Step 1

Rewrite the equation in vertex form.

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

Simplify $\frac{3}{4} y^{2}$ .

Rewrite.

$x - 7 = 0 + 0 + \frac{3}{4} y^{2}$

Simplify by adding zeros.

$x - 7 = \frac{3}{4} y^{2}$

Combine $\frac{3}{4}$ and $y^{2}$ .

$x - 7 = \frac{3 y^{2}}{4}$

Add $7$ to both sides of the equation.

$x = \frac{3 y^{2}}{4} + 7$

Complete the square for $\frac{3 y^{2}}{4} + 7$ .

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

$a = \frac{3}{4}$

$b = 0$

$c = 7$

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

$d = \frac{0}{2 (\frac{3}{4})}$

Simplify the right side.

Cancel the common factor of $0$ and $2$ .

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (\frac{3}{4})}$

Cancel the common factors.

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (\frac{3}{4})}$

Rewrite the expression.

$d = \frac{0}{\frac{3}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (\frac{4}{3})$

Multiply $0$ by $\frac{4}{3}$ .

$d = 0$

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

$e = 7 - \frac{0^{2}}{4 (\frac{3}{4})}$

Simplify the right side.

Simplify each term.

Raising $0$ to any positive power yields $0$ .

$e = 7 - \frac{0}{4 (\frac{3}{4})}$

Combine $4$ and $\frac{3}{4}$ .

$e = 7 - \frac{0}{\frac{4 \cdot 3}{4}}$

Multiply $4$ by $3$ .

$e = 7 - \frac{0}{\frac{12}{4}}$

Divide $12$ by $4$ .

$e = 7 - \frac{0}{3}$

Divide $0$ by $3$ .

$e = 7 - 0$

Multiply $- 1$ by $0$ .

$e = 7 + 0$

Add $7$ and $0$ .

$e = 7$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{3}{4} {(y + 0)}^{2} + 7$ .

$\frac{3}{4} {(y + 0)}^{2} + 7$

Set $x$ equal to the new right side.

$x = \frac{3}{4} \cdot {(y + 0)}^{2} + 7$

Step 2

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

$a = \frac{3}{4}$

$h = 7$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(7, 0)$

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.

$\frac{1}{4 \cdot \frac{3}{4}}$

Simplify.

Combine $4$ and $\frac{3}{4}$ .

$\frac{1}{\frac{4 \cdot 3}{4}}$

Simplify the expression.

Multiply $4$ by $3$ .

$\frac{1}{\frac{12}{4}}$

Divide $12$ by $4$ .

$\frac{1}{3}$

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.

$(\frac{22}{3}, 0)$

Step 6

Find the Focus x-7=3/4y^2

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