Question

write the standard form of the equation and the general form of the equation of the circle with radius r and center (h,k). then graph the circle.
r = 5; (h,k)=(4, - 3)
the standard form of the equation of this circle is (x - 4)^2+(y + 3)^2 = 25.
the general form of the equation of this circle is x^2 + y^2-2x = 0. (simplify your answer.)

Explanation:

Step1: Recall circle - standard - form formula

The standard - form of the equation of a circle with center \((h,k)\) and radius \(r\) is \((x - h)^2+(y - k)^2=r^2\).

Step2: Substitute given values

Given \(h = 4\), \(k=-3\), and \(r = 5\), we substitute into the formula: \((x - 4)^2+(y-(-3))^2=5^2\), which simplifies to \((x - 4)^2+(y + 3)^2=25\).

Step3: Expand to get general - form

Expand \((x - 4)^2+(y + 3)^2=25\).
\((x - 4)^2=x^{2}-8x + 16\) and \((y + 3)^2=y^{2}+6y+9\).
So \(x^{2}-8x + 16+y^{2}+6y + 9=25\).
Combine like - terms: \(x^{2}+y^{2}-8x + 6y+25 - 25=0\), which gives \(x^{2}+y^{2}-8x + 6y=0\).

Answer:

The standard form of the equation of the circle is \((x - 4)^2+(y + 3)^2=25\).
The general form of the equation of the circle is \(x^{2}+y^{2}-8x + 6y=0\).