Question

A circle with its center in the first quadrant touches both the coordinate axes and the line \( x - y - 2 = 0 \). Then the area of the circle is:

• A. \( 2\pi \)
• B. \( \pi \)
• C. \( \frac{\pi}{2} \)
• D. \( 4\pi \)

Hint:

When a circle touches both axes and a line, the distance from the center to the line gives the radius of the circle.

Answer 1

The Correct Option is A

The center of the circle lies in the first quadrant, meaning both the \( x \)- and \( y \)-coordinates of the center are positive. Let the center of the circle be \( (h, k) \). Since the circle touches the coordinate axes, the radius of the circle will be equal to \( h = k \), as the distance from the center to the \( x \)-axis and \( y \)-axis is equal to the radius. So, the equation of the circle will be: \[ (x - h)^2 + (y - k)^2 = h^2 \] The line \( x - y - 2 = 0 \) or \( x - y = 2 \) represents a line with slope 1. The distance from the center \( (h, k) \) to this line is given by the formula: \[ \text{Distance} = \frac{|h - k - 2|}{\sqrt{1^2 + (-1)^2}} = \frac{|h - h - 2|}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \] Since the circle touches the line, the distance from the center to the line is equal to the radius of the circle, \( h = \sqrt{2} \). Thus, the area of the circle is: \[ \text{Area} = \pi r^2 = \pi (\sqrt{2})^2 = 2\pi \] Therefore, the area of the circle is \( 2\pi \).