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 = 10; (h,k)=(-6,8)
the standard form of the equation of this circle is (x + 6)^2+(y - 8)^2 = 100.
the general form of the equation of this circle is x^2 + y^2+12x - 16y = 0. (simplify your answer.)
graph the circle.
click to enlarge graph

Explanation:

Step1: Recall circle standard - form formula

The standard - form of the equation of a circle is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center and $r$ is the radius. Given $h=-6$, $k = 8$, and $r = 10$, we substitute these values into the formula: $(x-(-6))^2+(y - 8)^2=10^2$, which simplifies to $(x + 6)^2+(y - 8)^2=100$.

Step2: Expand to get general - form

Expand $(x + 6)^2+(y - 8)^2=100$.
$(x + 6)^2=x^2+12x + 36$ and $(y - 8)^2=y^2-16y + 64$.
So $x^2+12x + 36+y^2-16y + 64=100$.
Combining like - terms: $x^2+y^2+12x-16y+100 = 100$.
Subtract 100 from both sides to get the general form $x^2+y^2+12x-16y=0$.

Answer:

The standard form of the equation of the circle is $(x + 6)^2+(y - 8)^2=100$.
The general form of the equation of the circle is $x^2+y^2+12x-16y=0$.