Question

Sketch the trace of the intersection of each plane with the given sphere. \(x^{2}+y^{2}+z^{2}-8 x-6 z+16=0\) (a) \(x=4\) (b) \(z=3\)

Short answer

The intersection of the sphere and the plane \(x=4\) is a circle centered at (0, 3) with radius 3. The intersection of the sphere and the plane \(z=3\) is a circle centered at (4, 0) with radius \( \sqrt{7}\).

Step-by-step solution

Substitute \(x = 4\) into the equation of sphere

The first step is to substitute \(x = 4\) into the equation of sphere, which gives:\[16 + y^{2} + z^{2} - 32 - 6z + 16 = 0.\]We simplify this to obtain:\[y^{2}+ z^{2} - 6z = 0\].

Solve for the intersection circle for \(x = 4\)

The resulting equation is of the form for a circle in two dimensions. Rewriting the equation to a standard circle form \(y^{2} + (z - k)^{2} = R^{2}\), we find that the circle of intersection between the sphere and the plane \(x = 4\) has a center at (0, 3) and a radius of 3.

Substitute \(z = 3\) into the equation of sphere

Next, substitute \(z = 3\) into the equation of the sphere, which gives:\[x^{2} + y^{2} + 9 - 8x - 18 + 16 = 0.\]We simplify this to obtain:\[x^{2} - 8x + y^{2} - 7 = 0\].

Solve for the intersection circle for \(z = 3\)

The resulting equation is also of the form for a circle in two dimensions. Therefore, by rearranging it to the standard circle form \(x^{2} + y^{2} = R^{2}\), the center and radius of the intersection circle between the sphere and the plane \(z = 3\) are obtained, being (4,0) and \( \sqrt{7}\), respectively.

Key concepts

Equation of a Sphere

The equation of a sphere in a three-dimensional space is typically expressed in a specific mathematical form. Understanding this is crucial to solving problems related to spheres, intersections, and geometry. The equation is given as \( (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2 \). Here, \( (h, k, l) \) is the center of the sphere, and \( r \) is its radius. This standard form conveys the geometric essence of a sphere, signifying all the points in space that are an equal distance (the radius) from the center.

In the problem, the given sphere's equation, \( x^2+y^2+z^2-8x-6z+16=0 \), needs to be manipulated into this standard form. By completing the square for the terms involving \( x, y, \) and \( z \), the problem becomes a bit more tangible. This manipulation reveals the center and allows for better geometric visualization, which assists in understanding the sphere's positioning in space.

Geometric Interpretation

A key skill in geometry is being able to visualize and interpret the equations of shapes like spheres and planes. This helps in comprehending their interactions, especially intersections.

• The equation \( x=4 \) represents a vertical plane parallel to the yz-plane, intersecting all points where \( x \) always equals 4.
• Similarly, \( z=3 \) represents a horizontal plane parallel to the xy-plane that cuts through all points where \( z \) always equals 3.

When you have a sphere, imagining these planes slice through it is like cutting an apple with a knife. Each plane creates a circular cross-section through the sphere. Understanding this framework lays a foundation for predicting intersection shapes without computation.

Circle of Intersection

When a plane intersects a sphere, the cross-section is a circle, assuming the plane isn't tangent to the sphere. Once you have the circle's equation, you can derive its center and radius, giving you comprehensive insight into the specific intersection.

Example: Plane \( x=4 \)

By plugging \( x=4 \) into the sphere's equation, we transform the three-dimensional problem into two dimensions, forming the equation \( y^{2} + (z-3)^2 = 3^2 \). This represents a circle in the yz-plane, centered at \( (0, 3) \) with radius 3.

Example: Plane \( z=3 \)

Inserting \( z=3 \), the problem reduces again to two dimensions: \( (x-4)^2 + y^{2} = ( ext{radius})^2 \). This gives the circle a center at \( (4, 0) \) and a radius of \( \sqrt{7} \).

These transformations from 3D to 2D highlight the regular shape—a circle—elucidating the intersection’s geometry and simplifying further analysis or graphical depiction.