Question

Multiple Choice Choose the equation of a circle with radius 6 and center (3,-5) (a) \((x-3)^{2}+(y+5)^{2}=6\) (b) \((x+3)^{2}+(y-5)^{2}=36\) (c) \((x+3)^{2}+(y-5)^{2}=6\) (d) \((x-3)^{2}+(y+5)^{2}=36\)

Short answer

The correct choice is (d) \( (x - 3)^2 + (y + 5)^2 = 36 \).

Step-by-step solution

Recall the Standard Form of a Circle

The standard form of the equation of a circle is \( (x-h)^2 + (y-k)^2 = r^2 \), where \( (h, k) \) is the center of the circle and \( r \) is the radius.

Identify Given Values

Given the center \( (3, -5) \) and radius 6, we can identify \( h = 3 \), \( k = -5 \), and \( r = 6 \).

Substitute Values into the Formula

Substitute \( h = 3 \), \( k = -5 \), and \( r = 6 \) into the equation \((x - h)^2 + (y - k)^2 = r^2\): \( (x - 3)^2 + (y - (-5))^2 = 6^2 \).

Simplify the Equation

Simplify the equation: \( (x - 3)^2 + (y + 5)^2 = 36 \).

Match with the Given Choices

Compare the simplified equation \( (x - 3)^2 + (y + 5)^2 = 36 \) with the given choices. The correct choice is (d).

Key concepts

Circle Equation Simplification

Sometimes the given equation might seem complex, but simplifying it using the standard form makes the task easier. For instance, we start with: \((x-3)^2 + (y-(-5))^2 = 6^2.\) Upon \begin{subarray}{l}Simplification,ewline (y-(-5)) converts to (y+5). So the complete equation: $ (x-3)^2 + (y+5)^2 = 36.ewlineewlineWhen comparing this to multiple-choice questions, look for the equation that matches the simplified form, ensuring the components align correctly. In our example, the right choice is (d).