Problem 19 - Essential Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

(a) \((x-2)^2 + (y+3)^2 + (z-6)^2 = 36\); (b) \((x-2)^2 + (y+3)^2 + (z-6)^2 = 4\); (c) \((x-2)^2 + (y+3)^2 + (z-6)^2 = 9\).

Step 1: Understanding the Sphere's Equation

The general equation of a sphere with center \((h, k, l)\) and radius \(r\) is given by:\[(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\]For the given sphere, the center is \((2, -3, 6)\). We need to find the equation by determining the radius in each case where the sphere touches a particular plane.

Step 2: Finding the Radius for Part (a) - Touches xy-plane

The distance from the center of the sphere \((2, -3, 6)\) to the \(xy\)-plane is the absolute value of the \(z\)-coordinate, which is 6. The radius of the sphere is this distance because the sphere just touches the plane, making:\(r = |6| = 6\).

Step 3: Writing the Equation for Part (a)

Based on the radius calculated, the equation becomes:\[(x - 2)^2 + (y + 3)^2 + (z - 6)^2 = 6^2\]This equation simplifies to:\[(x - 2)^2 + (y + 3)^2 + (z - 6)^2 = 36\]

Step 4: Finding the Radius for Part (b) - Touches yz-plane

The distance from the center \((2, -3, 6)\) to the \(yz\)-plane is the absolute value of the \(x\)-coordinate, which is 2. Therefore, the radius is:\(r = |2| = 2\).

Step 5: Writing the Equation for Part (b)

Using the radius found:\[(x - 2)^2 + (y + 3)^2 + (z - 6)^2 = 2^2\]This simplifies the equation to:\[(x - 2)^2 + (y + 3)^2 + (z - 6)^2 = 4\]

Step 6: Finding the Radius for Part (c) - Touches xz-plane

The distance from the center \((2, -3, 6)\) to the \(xz\)-plane is the absolute value of the \(y\)-coordinate, which is 3. So, the radius \(r\) is:\(r = |-3| = 3\).

Step 7: Writing the Equation for Part (c)

Using the radius calculated:\[(x - 2)^2 + (y + 3)^2 + (z - 6)^2 = 3^2\]This results in the equation:\[(x - 2)^2 + (y + 3)^2 + (z - 6)^2 = 9\]

Key Concepts

Center of Sphere

The center of a sphere is a crucial component when it comes to writing the equation of a sphere in three-dimensional space. It is denoted typically by the coordinates \(h, k, l\). Think of it as the "origin" point from which every other point on the surface of the sphere is equidistant. In the context of the given exercise, the center is \(2, -3, 6\).

This means if you were standing at point \(2, -3, 6\), you would be at the exact middle part of the sphere. Everything else (the radius and the sphere's surface) radiates from this point. The general equation of a sphere, \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\), is centered around these coordinates.

Understanding how to pinpoint and use the sphere's center helps to simplify calculations, especially when a sphere interacts with planes, like touching them in this exercise.

Radius of Sphere

The radius of a sphere is the distance from the center of the sphere to any point on its surface. In mathematical terms, it is represented by the variable \(r\) in the equation of the sphere \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\)\. For spheres touching planes, this radius is the exact distance from the center to the closest point on the plane.

In our scenario, determining the radius involves finding the distance from the center \(2, -3, 6\) to each plane:

• To the \(\textit{xy}\)-plane: The distance is the absolute value of the \(z\)-coordinate (6), making the radius 6.
• To the \(\textit{yz}\)-plane: Calculated using the absolute value of the \(x\)-coordinate (2), so the radius is 2.
• To the \(\textit{xz}\)-plane: This is the absolute \text{value of the } \ \(y\)-coordinate (-3), hence, the radius is 3.

Recognizing the radius is directly linked to these coordinates makes it easy to decide how far the sphere extends to meet the plane's surface.

Touching Planes

When a sphere "touches" a plane, it means the sphere is tangent to the plane at exactly one point. This condition is key when determining the radius used in the sphere’s equation. In this situation, the radius of the sphere equals the distance from its center to the plane.

For the different planes involved in this exercise:

• The \(\textit{xy}\)-plane touches the z-coordinate of the center, meaning the sphere extends vertically upward to this plane.
• The \(\textit{yz}\)-plane touches the x-coordinate of the center, so the sphere extends horizontally toward this plane.
• The \(\textit{xz}\)-plane is tangent to the y-coordinate, causing the sphere to stretch horizontally to this plane as well.

In each case, the point of contact is directly above or below (or to the side) of the center along the axis not part of the plane, affirming where the sphere makes its brush against the plane.

Distance to a Plane

The concept of distance to a plane is central to solving how a sphere interacts with a plane. For a sphere with a center at any point, calculating the distance to a plane involves checking the absolute coordinate not shared by that plane.

• For the \(\textit{xy}\)-plane, it's the vertical distance given by the \(z\)-coordinate. If the center is \(2, -3, 6\), this distance is simply 6.
• For the \(\textit{yz}\)-plane, it’s determined by the horizontal distance along the \(x\)-axis, so it would be 2 as per center \(2, -3, 6\).
• For the \(\textit{xz}\)-plane, the relevant distance is the value along the \(y\)-axis, in this instance, \|-3|\ or 3.

Understanding this connection allows for quick recalculation of an equation when adjusting any sphere's position relative to various planes.