Discover a world of knowledge and get your questions answered at IDNLearn.com. Find reliable solutions to your questions quickly and easily with help from our experienced experts.

Find the center of the ellipse.

[tex]\[ 25x^2 + y^2 - 100x - 2y + 76 = 0 \][/tex]


Sagot :

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].