To rewrite the given equation of the circle in standard form and determine its center and radius, follow these steps:
1. Given Equation:
The initial form of the equation is [tex]\(7x^2 + 7y^2 - 28x + 42y - 35 = 0\)[/tex].
2. First Simplification:
Divide the entire equation by 7 to simplify:
[tex]\[
x^2 + y^2 - 4x + 6y - 5 = 0
\][/tex]
3. Completing the Square for the x-terms:
Group the x terms together: [tex]\(x^2 - 4x\)[/tex].
- Complete the square: [tex]\((x^2 - 4x)\)[/tex] can be written as [tex]\((x - 2)^2 - 4\)[/tex].
4. Completing the Square for the y-terms:
Group the y terms together: [tex]\(y^2 + 6y\)[/tex].
- Complete the square: [tex]\((y^2 + 6y)\)[/tex] can be written as [tex]\((y + 3)^2 - 9\)[/tex].
5. Writing the Completed Squares in the Equation:
Substitute the completed squares back into the equation:
[tex]\[
(x - 2)^2 - 4 + (y + 3)^2 - 9 = 5
\][/tex]
6. Simplifying:
Combine the constants on the right-hand side:
[tex]\[
(x - 2)^2 + (y + 3)^2 - 13 = 5
\][/tex]
7. Final Simplification:
Move the -13 to the right-hand side to form the standard equation of a circle:
[tex]\[
(x - 2)^2 + (y + 3)^2 = 18
\][/tex]
8. Center and Radius:
- The standard form equation is [tex]\((x - 2)^2 + (y + 3)^2 = 18\)[/tex].
- The center of the circle, [tex]\((h, k)\)[/tex], is [tex]\((2, -3)\)[/tex].
- The radius [tex]\(r\)[/tex] is [tex]\(\sqrt{18}\)[/tex], which simplifies to approximately 4.2426.
So, filling in the blanks:
The general form of the equation of a circle is [tex]\(7 x^2+7 y^2-28 x+42 y-35=0\)[/tex].
The equation of this circle in standard form is [tex]\((x - 2)^2 + (y + 3)^2 = 18\)[/tex]. The center of the circle is at the point [tex]\((2, -3)\)[/tex], and its radius is approximately 4.2426 units.
