Problem 50 - College Algebra and Calculus: An Applied Approach

Question

Sketch the \(x y\) -trace of the sphere. $$ x^{2}+y^{2}+z^{2}-4 y+2 z-60=0 $$

Step-by-Step Solution

Answer

The \(xy\)-trace of the sphere is a circle with a radius of 8 and its center at the point (0,2).

Step 1: Set z to zero

The equation of the sphere was given as \(x^{2}+y^{2}+z^{2}-4 y+2 z-60=0\). In order to find the \(xy\)-trace, set \(z=0\). This gives the new equation: \(x^{2}+y^{2}-4 y-60=0\)

Step 2: Continue to complete the square

Completing the square gives a clearer view of the properties of the circle. Rewrite the equation as follows: \(x^{2}+(y-2)^{2}-64=0\)

Step 3: Write the equation in standard form

The equation \(x^{2}+(y-2)^{2}-64=0\) can be rewritten into the standard form of a circle \(x^{2}+(y-\delta)^{2}=\rho^{2}\), where \(\delta\) is the y-value of the center of the circle and \(\rho\) is the radius. This gives \((x-0)^{2}+(y-2)^{2}=8^{2}\), which has the center at the point (0,2) and the radius as 8.

Step 4: Sketch the circle

Now draw a circle on the \(xy\)-plane. The center of the circle must be at (0,2) and the radius of the circle will be 8 units. This will be the \(xy\)-trace of the sphere.