Consider again the sphere of radius 5 centered at (2, 3,−6). Describe the intersection of the sphere with the xz-plane. Write an equation (or equations) for this curve.

Reiny, I can't

Well, if we're talking about the intersection of the sphere with the xz-plane, it means we're only interested in the points where the y-coordinate is zero. So, let's set the y-coordinate to zero and see what we get!

The equation of the sphere is (x - 2)^2 + (y - 3)^2 + (z + 6)^2 = 25. If we set y to zero, we get (x - 2)^2 + 9 + (z + 6)^2 = 25, which simplifies to (x - 2)^2 + (z + 6)^2 = 16.

So, the equation for the curve formed by the intersection of the sphere with the xz-plane is (x - 2)^2 + (z + 6)^2 = 16. But hey, if you ask me, this equation is quite roundabout! 😄 Get it? Because it's a sphere intersecting a plane? 🤡

To describe the intersection of the sphere with the xz-plane, we need to find the points where the sphere intersects the xz-plane.

First, let's recall the equation of a sphere in 3D space centered at (a, b, c) with radius r:
(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2

In this case, the sphere is centered at (2, 3, -6) with a radius of 5. So the equation of the sphere is:
(x - 2)^2 + (y - 3)^2 + (z + 6)^2 = 5^2

To find the intersection with the xz-plane, we set y to zero, since points on the xz-plane have a y-coordinate of zero. Therefore, we can simplify the equation by substituting y with 0:
(x - 2)^2 + (0 - 3)^2 + (z + 6)^2 = 5^2
(x - 2)^2 + 9 + (z + 6)^2 = 25

Now, we can further simplify the equation to obtain the equation of the curve in the xz-plane:
(x - 2)^2 + (z + 6)^2 = 16

This equation represents the intersection of the sphere with the xz-plane.

To describe the intersection of the sphere with the xz-plane, we need to determine the set of points on the sphere that lie in the xz-plane.

The xz-plane is a plane where the y-coordinate is always 0. In other words, any point in the xz-plane will have the form (x, 0, z).

The equation of the sphere is given by:
(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2

Where:
(a, b, c) is the center of the sphere.
r is the radius of the sphere.

In this case, the center of the sphere is (2, 3, -6) and the radius is 5. So, the equation of the sphere becomes:
(x - 2)^2 + (y - 3)^2 + (z + 6)^2 = 5^2

Since we are looking for the intersection with the xz-plane, we can set the y-coordinate to 0 in the equation of the sphere:
(x - 2)^2 + (0 - 3)^2 + (z + 6)^2 = 5^2
(x - 2)^2 + (-3)^2 + (z + 6)^2 = 25
(x - 2)^2 + 9 + (z + 6)^2 = 25
(x - 2)^2 + (z + 6)^2 = 16

Therefore, the equation of the curve representing the intersection of the sphere with the xz-plane is:
(x - 2)^2 + (z + 6)^2 = 16

Isn't the equation

(x-2)^2 + (y-3)^2 + (z+6)^2 = 25

And on the x-z plane, isn't y = 0 ?
so
(x-2)^2 + (0-3)^2 + (z+6)^2 = 25
(x-2)^2 + (z+6)^2 = 25 + 9
(x-2)^2 + (z+6)^2 = 34

take it from there.