The center of circle Q has coordinates `(3, -2)` in the xy-plane. If an endpoint of a radius of circle Q has coordinates `R(7, 1)`, what is the equation of circle Q?

Options

A. `(x-3)^(2)+(y+2)^(2)=5`
B. `(x+3)^(2)+(y-2)^(2)=25`
C. `(x-3)^(2)+(y+2)^(2)=25`
D. `(x+3)^(2)+(y-2)^(2)=5`

Answer

To find the equation of circle Q with center at coordinates \( (3, -2) \) and an endpoint of a radius at coordinates \( R(7, 1) \), we will follow these steps:

### Step 1: Identify the center and the endpoint of the radius
The center of the circle \( Q \) is given as \( (3, -2) \) and the endpoint of the radius \( R \) is \( (7, 1) \).

### Step 2: Use the distance formula to find the radius
The radius \( r \) can be calculated using the distance formula, which is given by:

\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Here, \( (x_1, y_1) = (3, -2) \) (the center) and \( (x_2, y_2) = (7, 1) \) (the endpoint of the radius).

Substituting the values into the formula:

\[ r = \sqrt{(7 - 3)^2 + (1 - (-2))^2} \]

### Step 3: Simplify the expression
Calculating the differences:

\[ r = \sqrt{(4)^2 + (1 + 2)^2} \]

This simplifies to:

\[ r = \sqrt{16 + 9} \]

### Step 4: Calculate the radius
Now, we can calculate \( r \):

\[ r = \sqrt{25} = 5 \]

### Step 5: Write the equation of the circle
The standard equation of a circle with center \( (h, k) \) and radius \( r \) is given by:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

Substituting \( h = 3 \), \( k = -2 \), and \( r = 5 \):

\[ (x - 3)^2 + (y + 2)^2 = 5^2 \]

### Step 6: Finalize the equation
This simplifies to:

\[ (x - 3)^2 + (y + 2)^2 = 25 \]

Thus, the equation of circle Q is:

\[ \boxed{(x - 3)^2 + (y + 2)^2 = 25} \]