Quadric Surfaces — Question 9

Question 9

Find and classify the curve of intersection of the sphere $x^2+y^2+z^2=25$ with the plane $x+2y+2z=9$. Find the curve’s center and radius, and determine whether the plane is tangent, secant, or disjoint.

Question 9 – Solution

Strategy The circle center is the perpendicular projection of the sphere center onto the plane.

Plane distance The origin-to-plane distance is $d=9/\sqrt{1+4+4}=3$. Since $3<5$, the plane is secant.

Center and radius The foot from the origin lies along $\left\langle 1,2,2\right\rangle$ and is $$C=\frac{9}{9}\left\langle 1,2,2\right\rangle=\boxed{(1,2,2)}.$$ By the right triangle from sphere center to circle edge, $r^2=5^2-3^2=16$, so $\boxed{r=4}$.

Classification The intersection is a circle in the given plane, centered at $(1,2,2)$ with radius 4.