Problem 67 - College Algebra

Question

Does \((x-3)^{2}+(y-5)^{2}=-25\) represent the equation of a circle? What sort of set is the graph of this equation?

Step-by-Step Solution

Answer

The equation does not represent a circle. The graph of this equation is the empty set as the equation has no real solutions.

Step 1: Comparing with Standard Equation

Comparing our given equation with the standard circle equation, we have \((a,b) = (3,5)\), and \(r^{2} = -25\).

Step 2: Checking for Validity of Radius

For the equation to represent a circle, the radius must be real and non-negative. Here, the square of the radius is -25. But squares of real numbers are always non-negative.

Step 3: Concluding on Type of Set

Since the squared radius in our equation is negative, which cannot be true for any real radius, the equation cannot represent a real circle. Therefore, it has no graph in the real plane, and the set of solutions to this equation is the empty set.