Find the Focus x^2-2x+y-1=0

$x^{2} - 2 x + y - 1 = 0$

Step 1

Rewrite the equation in vertex form.

Move all terms not containing $y$ to the right side of the equation.

Subtract $x^{2}$ from both sides of the equation.

$- 2 x + y - 1 = - x^{2}$

Add $2 x$ to both sides of the equation.

$y - 1 = - x^{2} + 2 x$

Add $1$ to both sides of the equation.

$y = - x^{2} + 2 x + 1$

Complete the square for $- x^{2} + 2 x + 1$ .

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

$a = - 1$

$b = 2$

$c = 1$

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

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

Simplify the right side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$d = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$d = - 1$

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

$e = 1 - \frac{2^{2}}{4 \cdot - 1}$

Simplify the right side.

Simplify each term.

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

$e = 1 - \frac{4}{4 \cdot - 1}$

Multiply $4$ by $- 1$ .

$e = 1 - \frac{4}{- 4}$

Divide $4$ by $- 4$ .

$e = 1 - - 1$

Multiply $- 1$ by $- 1$ .

$e = 1 + 1$

Add $1$ and $1$ .

$e = 2$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - 1)}^{2} + 2$ .

$- {(x - 1)}^{2} + 2$

Set $y$ equal to the new right side.

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

Step 2

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

$a = - 1$

$h = 1$

$k = 2$

Step 3

Find the vertex $(h, k)$ .

$(1, 2)$

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.

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

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

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

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

Move the negative in front of the fraction.

$- \frac{1}{4}$

Step 5

Find the focus.

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

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

$(1, \frac{7}{4})$

Step 6

Find the Focus x^2-2x+y-1=0

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