Question

write the standard form of the equation and the general form of the equation of the circle of radius r and center (h,k). graph the circle. r = \sqrt{7}; (h,k)=(4, - 2) the equation for the circle in standard form is (x - 4)^2+(y + 2)^2 = 7. (simplify your answer.) the equation for the circle in general form is . (simplify your answer.)

Explanation:

Step1: Expand the standard - form equation

We start with the standard - form equation \((x - 4)^2+(y + 2)^2=7\). Expand \((x - 4)^2\) and \((y + 2)^2\) using the formula \((a\pm b)^2=a^{2}\pm2ab + b^{2}\).
\((x - 4)^2=x^{2}-8x + 16\) and \((y + 2)^2=y^{2}+4y+4\).
So, \(x^{2}-8x + 16+y^{2}+4y + 4=7\).

Step2: Rearrange to general form

Combine like - terms and set the equation equal to 0.
\(x^{2}+y^{2}-8x + 4y+16 + 4-7=0\).
\(x^{2}+y^{2}-8x + 4y+13=0\).

Answer:

\(x^{2}+y^{2}-8x + 4y+13 = 0\)