To find the center of the ellipse given by the equation [tex]\(25x^2 + y^2 - 100x - 2y + 76 = 0\)[/tex], we need to complete the square for both the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] terms. Follow these steps:
1. Group and factor out coefficients of the quadratic terms:
- Group the terms involving [tex]\(x\)[/tex]:
[tex]\[
25x^2 - 100x
\][/tex]
and the terms involving [tex]\(y\)[/tex]:
[tex]\[
y^2 - 2y
\][/tex]
2. Complete the square for [tex]\(x\)[/tex]-terms:
- Factor out the coefficient of [tex]\(x^2\)[/tex], which is 25, from the [tex]\(x\)[/tex]-terms:
[tex]\[
25(x^2 - 4x)
\][/tex]
- To complete the square inside the parentheses:
[tex]\[
x^2 - 4x = (x - 2)^2 - 4
\][/tex]
- Substituting back:
[tex]\[
25(x^2 - 4x) = 25((x - 2)^2 - 4) = 25(x - 2)^2 - 100
\][/tex]
3. Complete the square for [tex]\(y\)[/tex]-terms:
- Take the [tex]\(y\)[/tex]-terms:
[tex]\[
y^2 - 2y
\][/tex]
- Complete the square:
[tex]\[
y^2 - 2y = (y - 1)^2 - 1
\][/tex]
4. Rewrite the original equation using these perfect squares:
- Substitute the completed squares back into the equation:
[tex]\[
25(x - 2)^2 - 100 + (y - 1)^2 - 1 + 76 = 0
\][/tex]
- Combine the constants:
[tex]\[
25(x - 2)^2 + (y - 1)^2 - 100 - 1 + 76 = 0
\][/tex]
[tex]\[
25(x - 2)^2 + (y - 1)^2 - 25 = 0
\][/tex]
5. Simplify the equation:
[tex]\[
25(x - 2)^2 + (y - 1)^2 - 25 = 0
\][/tex]
To isolate the completed squares:
[tex]\[
25(x - 2)^2 + (y - 1)^2 = 25
\][/tex]
6. Identify the center of the ellipse:
- The standard form of the ellipse equation after completing the square is:
[tex]\[
25(x - 2)^2 + (y - 1)^2 = 25
\][/tex]
- From this form, it is clear that the ellipse is centered at [tex]\((x, y) = (2, 1)\)[/tex].
Therefore, the center of the ellipse is [tex]\((2, 1)\)[/tex].
