Question

A circle with center (-2,1) is tangent to the y-axis. What is the radius, equation, and is the circle also tangent to the x-axis?

Answer 1

The radius of the circle tangent to the y-axis with center (-2,1) is 2. The equation of the circle is (x + 2)^2 + (y - 1)^2 = 4, and it is not tangent to the x-axis.

Step-by-step explanation:

The student's question pertains to finding the radius and equation of a circle that is tangent to the y-axis, with a given center at (-2,1), and determining whether the circle is also tangent to the x-axis.

Steps to Solve the Problem

1. Identify the knowns: center of the circle (-2,1) and that the circle is tangent to the y-axis.
2. Since the circle is tangent to the y-axis, the distance from the center to the y-axis gives us the radius. The x-coordinate of the center is -2, thus the radius is 2.
3. The general formula for a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the center and r is the radius.
4. Plugging in the known values, we get the circle's equation: (x + 2)^2 + (y - 1)^2 = 2^2.
5. To determine if the circle is also tangent to the x-axis, we can check if the distance from the center to the x-axis equals the radius. Since the y-coordinate of the center is 1, and the radius is 2, the circle cannot be tangent to the x-axis as it is not at a distance of 2 from it.

We have determined that the radius is 2, the equation of the circle is (x + 2)^2 + (y - 1)^2 = 4, and the circle is not tangent to the x-axis.