To find the focus of the parabola given by the equation [tex]\( y^2 - 2y = 4x + 3 \)[/tex], we need to rewrite it in a standard form and use it to determine the focus.
1. Start with the given equation:
[tex]\[
y^2 - 2y = 4x + 3
\][/tex]
2. Rearrange the equation to make it easier to complete the square:
[tex]\[
y^2 - 2y = 4x + 3
\][/tex]
Subtract 4x and 3 from both sides:
[tex]\[
y^2 - 2y - 4x - 3 = 0
\][/tex]
3. Complete the square for the [tex]\( y \)[/tex]-terms:
- Identify the coefficient of [tex]\( y \)[/tex], which is -2.
- Divide -2 by 2, which gives -1.
- Square -1 to get 1.
So, we add and subtract 1 inside the equation:
[tex]\[
y^2 - 2y + 1 - 1 - 4x - 3 = 0
\][/tex]
[tex]\[
(y - 1)^2 - 1 - 4x - 3 = 0
\][/tex]
4. Simplify the equation:
[tex]\[
(y - 1)^2 - 4x - 4 = 0
\][/tex]
Add 4 to both sides to isolate the completed square:
[tex]\[
(y - 1)^2 = 4x + 4
\][/tex]
Rewrite it:
[tex]\[
(y - 1)^2 = 4(x + 1)
\][/tex]
5. Express it in the standard form of a parabola:
The form [tex]\((y - k)^2 = 4p(x - h)\)[/tex] represents a parabola that opens to the right if [tex]\( p > 0 \)[/tex].
6. Identify the parameters:
- [tex]\( h = -1 \)[/tex]
- [tex]\( k = 1 \)[/tex]
- [tex]\( 4p = 4 \)[/tex] implies that [tex]\( p = 1 \)[/tex]
7. Find the focus:
The focus [tex]\((h + p, k)\)[/tex] of the parabola is:
[tex]\[
(-1 + 1, 1) = (0, 1)
\][/tex]
Therefore, the focus of the parabola [tex]\(y^2 - 2y = 4x + 3\)[/tex] is at:
[tex]\[
\boxed{(0, 1)}
\][/tex]
