Quadric Surfaces — Question 6

Question 6

Find and describe the intersection of $$x^2+y^2+z^2=25\quad\text{and}\quad z^2=x^2+y^2.$$

Tasks

1. Determine every component of the intersection.

2. Find each component’s exact height and radius.

3. Verify the result in both original equations.

Question 6 – Solution

Strategy Substitute the cone relation into the sphere, then retain both signs of $z$.

Step 1: Solve for height Since $x^2+y^2=z^2$, the sphere gives $$z^2+z^2=25\Longrightarrow 2z^2=25\Longrightarrow z=\pm\frac 5{\sqrt 2}.$$

Step 2: Solve for radius On either level, $$x^2+y^2=z^2=\frac{25}{2}.$$ Thus the intersection consists of two circles: $$\boxed{z=\pm\frac 5{\sqrt 2},\qquad x^2+y^2=\frac{25}{2}}.$$ Each has radius $\boxed{5/\sqrt 2}$. Both signs are essential because the cone is double. Substitution verifies both circles in both original equations.