Question

Sketch the trace of the intersection of each plane with the given sphere. \(x^{2}+y^{2}+z^{2}-4 x-6 y+9=0\) (a) \(x=2\) (b) \(y=3\)

Short answer

The intersection of the sphere with the plane \(x=2\) is a circle of radius 4 centered at (3,0) on the yz plane. The intersection of the sphere with the plane \(y=3\) is a circle of radius 4 centered at (2,0) on the xz plane.

Step-by-step solution

Rearrange the Sphere's Equation

We want to get the equation of the sphere into a more standard form. By rearranging the terms, we get \(x^{2}-4x+y^{2}-6y+z^{2}=9\). If we complete the square for x and y terms, the equation becomes \((x-2)^{2}+(y-3)^{2}+z^{2}=16\). Now it's easy to see the center of the sphere is at the point (2,3,0) with radius 4.

Finding the Intersection with the Plane \(x=2\)

Now we substitute \(x=2\) into the rearranged sphere equation. This simplifies to \(0+(y-3)^{2}+z^{2}=16 \) or \((y-3)^{2}+z^{2}=16\). This implies a circle centered at (3,0) on the yz-plane with radius 4.

Finding the Intersection with the Plane \(y=3\)

Now we substitute \(y=3\) into the rearranged sphere equation. This simplifies to \((x-2)^{2}+0+z^{2}=16 \) or \((x-2)^{2}+z^{2}=16\). This implies a circle centered at (2,0) on the xz-plane with radius 4.

Key concepts

Equation of a Sphere

The equation of a sphere is a way to describe all points in space that are equidistant from a center point. It typically takes the form \[ (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2 \] where

• \( (h, k, l) \) is the center of the sphere.
• \( r \) is the radius.

In the original problem, we start with \[ x^2 + y^2 + z^2 - 4x - 6y + 9 = 0. \] To get this into the standard form, we rearrange and complete the square, which we'll explore in another section. This gives the equation \[ (x-2)^2 + (y-3)^2 + z^2 = 16, \] indicating a sphere centered at \((2, 3, 0)\) with radius 4.
Understanding this format makes it easier to visualize the sphere in space.

Plane Intersection

When a plane intersects a sphere, the intersection is generally a circle. To find where a specific plane cuts the sphere, we substitute the plane's equation into the sphere's equation.
For the plane \( x=2 \), substituting into the sphere gives: \[ (y-3)^2 + z^2 = 16. \] This shows a circle centered at \((3, 0)\) on the \(yz\)-plane with radius 4.
For the plane \( y=3 \), substituting gives: \[ (x-2)^2 + z^2 = 16. \] This circle is centered at \((2, 0)\) on the \(xz\)-plane with radius 4.
This method allows us to clearly see the trace that each plane makes as it slices through the sphere.

Completing the Square

Completing the square is a technique used to convert a quadratic expression into a squared binomial form. It is especially useful in geometry to transform equations into standard geometric forms.
To complete the square in the sphere's equation \[ x^2 - 4x + y^2 - 6y + z^2 = 9, \] we adjust the \(x\) and \(y\) terms:

• For \(x\), calculate \( (x-2)^2 = x^2 - 4x + 4 \). Add 4 to both sides.
• For \(y\), calculate \( (y-3)^2 = y^2 - 6y + 9 \). Add 9 to both sides.

The equation becomes \[ (x-2)^2 + (y-3)^2 + z^2 = 16. \]
Completing the square reveals the sphere's true geometric form, simplifying further calculations.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is a branch of mathematics that uses algebra to study geometric shapes. By representing geometric figures using coordinates and equations, we can solve various problems algebraically.
The sphere's equation given in coordinate form allows us to find points in space and understand intersections with other surfaces, like planes.

• This helps visualize objects in three dimensions.
• It provides a systematic approach to solving spatial problems.

By understanding how coordinates define shapes, we can easily interpret the effects of transformations and translate between algebraic and geometric representations.