A circle in the xy-plane is tangent to the x-axis at -10 and the y-axis at 10. Which of the following is an equation of the circle ?

Options

A. (x-10)+(y+10)=100
B. `(x-10)^2 + (y+10)^2 =100`
C. `(x-10)^2 + (y-10)^2 =100`
D. `(x+10)^2 + (y-10)^2 =100`

Answer

To find the equation of the circle that is tangent to the x-axis at (-10, 0) and the y-axis at (0, 10), we can follow these steps:

### Step 1: Identify the points of tangency
The circle is tangent to the x-axis at the point (-10, 0) and to the y-axis at the point (0, 10).

### Step 2: Determine the center of the circle
Since the circle is tangent to the x-axis at (-10, 0), the center of the circle must be directly above this point. Similarly, since it is tangent to the y-axis at (0, 10), the center must be directly to the left of this point. Thus, the center of the circle is at the coordinates (-10, 10).

### Step 3: Calculate the radius of the circle
The radius of the circle can be determined by the distance from the center to either of the points of tangency. The distance from the center (-10, 10) to the point of tangency on the x-axis (-10, 0) is:
\[ r = |10 - 0| = 10 \]
Alternatively, the distance from the center (-10, 10) to the point of tangency on the y-axis (0, 10) is:
\[ r = |(-10) - 0| = 10 \]
Thus, the radius \( r \) is 10.

### Step 4: Write the equation of the circle
The general equation of a circle with center (a, b) and radius r is given by:
\[ (x - a)^2 + (y - b)^2 = r^2 \]
Substituting the center (-10, 10) and the radius 10 into the equation:
\[ (x - (-10))^2 + (y - 10)^2 = 10^2 \]
This simplifies to:
\[ (x + 10)^2 + (y - 10)^2 = 100 \]

### Step 5: Finalize the equation
Thus, the equation of the circle is:
\[ (x + 10)^2 + (y - 10)^2 = 100 \]

### Conclusion
The correct equation of the circle is:
\[ (x + 10)^2 + (y - 10)^2 = 100 \]