Find equations of the spheres with center (2,-3,6) that touch (a) the x y-plane, (b) the y z-pla

step 1 Let's find some equations for some spheres. All of these spheres are going to have their center

Find equations of the spheres with center (2,-3,6) that...

Key Concepts

Equation of a Sphere

The equation of a sphere in three-dimensional space is given by (x - a)^2 + (y - b)^2 + (z - c)^2 = r^2, where (a, b, c) is the center of the sphere and r is its radius. This standard form allows us to easily identify the location and size of the sphere.

Distance from a Point to a Plane

The distance from a point to a plane is the shortest distance along a line perpendicular to the plane. For example, for the xy-plane which has the equation z = 0, the distance from a point (x, y, z) to the plane is |z|. This concept is used to determine the radius of a sphere that is tangent to a plane.

Tangency Condition to a Plane

A sphere is tangent to a plane if the distance from its center to the plane is equal to its radius. By calculating the absolute value of the appropriate coordinate of the center corresponding to the plane (e.g., |z| for the xy-plane, |x| for the yz-plane, |y| for the xz-plane), we can determine the exact radius needed for the sphere to just touch the plane.

Transcript

00:03 Let's find some equations for some spheres.

00:06 All of these spheres are going to have their center at 2, negative 3, 6.

00:12 The first one's going to be just touching the x, y, y, plane at a single point.

00:18 The next one's going to be touching the y, z plane at a single point, and the last one's going to be touching the x, z plane at a single point.

00:28 To write an equation for a sphere, we need the coordinates of the center, and the length of the radius.

00:35 We have the coordinates for the center, which is in fact going to be the same for all three of these.

00:40 So all three of these are going to start off, x minus 2 squared, plus y minus negative 3 squared, plus z minus 6 squared, just a matter of finding out what the radius is for each one.

00:57 So if i want to find the radius for the xy plane, the sphere that touches the xy plane, i need to know how far the center is from the xy plane.

01:18 Well, the distance a point is from the xy plane is going to be given by its z coordinate.

01:27 And so the z coordinate of six tells us that this point is six units above the xy plane.

01:36 So that has to be the radius.

01:39 That's its distance to the end of the sphere.

01:42 And so we're looking for a circle with radius 6 squared.

01:49 Now i'm going to simplify this by changing the subtraction of a negative 3 to to y plus 3.

01:57 And i'm going to square the 6.

01:59 So i can get x minus 2 squared plus y plus 3 squared plus z minus 6 squared equals 36.

02:14 For the one that's just touching the yz plane, the distance of point is from the yz plane is given by its x coordinate...

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