Answer
a. center $(0, 2, -10)$, radius $3\sqrt{11}$
b. center $(4, 2, -10)$, radius $5\sqrt{3}$
Work Step by Step
$(x+1)^2+(y-2)^2+(z+10)^2=100$
a. At any point on the $yz$-plane, $x=0$. So to get the trace of the sphere in the $yz$-plane we can simply plug in $x=0$. The equation becomes:
$(0+1)^2+(y-2)^2+(z+10)^2=100$
$1^2+(y-2)^2+(z+10)^2=100$
$1+(y-2)^2+(z+10)^2=100$
$(y-2)^2+(z+10)^2=99$ (with $x=0$)
This is a circle in the $yz$-plane centered at $(0, 2, -10)$ with radius $\sqrt{99}=3\sqrt{11}$.
b. To get the trace of the sphere in the plane $x=4$ we can simply plug in $x=4$. The equation becomes:
$(4+1)^2+(y-2)^2+(z+10)^2=100$
$5^2+(y-2)^2+(z+10)^2=100$
$25+(y-2)^2+(z+10)^2=100$
$(y-2)^2+(z+10)^2=75$ (with $x=4$)
This is a circle in the $yz$-plane centered at $(4, 2, -10)$ with radius $\sqrt{75}=5\sqrt{3}$.
