The center of circle O (not shown) falls on the point where the line `y=(4)/(3)x+4` intersects the x-axis on the coordinate plane. The point (3, 8) lies on the circumference of the circle. Which of the following could be the equation for circle O?

Options

A. `x^(2)+y^(2)=25`
B. `(x+3)^(2)+y^(2)=25`
C. `(x+3)^(2)+y^(2)=100`
D. `(x+3)^(2)+(y-8)^(2)=100`

Answer

To solve the problem step by step, we need to find the equation of circle O given that its center lies at the intersection of the line \( y = \frac{4}{3}x + 4 \) and the x-axis, and that the point (3, 8) lies on the circumference of the circle.

### Step 1: Find the intersection point of the line and the x-axis.
The x-axis is represented by the equation \( y = 0 \). To find the intersection point, we set \( y \) in the line equation to 0:

\[ 0 = \frac{4}{3}x + 4 \]

### Step 2: Solve for \( x \).
Rearranging the equation gives:

\[ \frac{4}{3}x = -4 \]

Multiplying both sides by \( \frac{3}{4} \):

\[ x = -3 \]

### Step 3: Determine the coordinates of the center of the circle.
Since \( y = 0 \) at the intersection, the coordinates of the center \( O \) are:

\[ (-3, 0) \]

### Step 4: Calculate the radius of the circle.
The radius is the distance from the center \( O(-3, 0) \) to the point on the circumference \( P(3, 8) \). We use the distance formula:

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

Substituting the coordinates:

\[ d = \sqrt{(3 - (-3))^2 + (8 - 0)^2} \]

### Step 5: Simplify the expression.
Calculating inside the square root:

\[ d = \sqrt{(3 + 3)^2 + (8)^2} = \sqrt{(6)^2 + (8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]

Thus, the radius \( r \) is 10.

### Step 6: Write the equation of the circle.
The general equation of a circle with center \( (h, k) \) and radius \( r \) is:

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

Substituting \( h = -3 \), \( k = 0 \), and \( r = 10 \):

\[ (x + 3)^2 + (y - 0)^2 = 10^2 \]

This simplifies to:

\[ (x + 3)^2 + y^2 = 100 \]

### Final Equation:
The equation of circle O is:

\[ (x + 3)^2 + y^2 = 100 \]