Determining the Center and Radius of a Tangent Circle

The problem discusses the relationship between a line and a circle, specifically focusing on finding the center and radius of a circle given certain conditions. This article will explore the solution step-by-step, providing a clear understanding of the geometric principles involved.

Problem Description

A circle's center is located on the line y 2x, and the line x 1 is tangent to the circle at the point (1, 6). The goal is to find the center and radius of the circle.

Solution

Step 1: Identify the given information and setup equations.

The center of the circle is on the line y 2x. The line x 1 is tangent to the circle at the point (1, 6). The normal to the tangent at the point (1, 6) is the vertical line x 1.

Step 2: Determine the center of the circle.

Since the center lies on the line y 2x, we can express the coordinates of the center as (h, k) where k 2h.

The tangent point is (1, 6), and the normal line at this point is vertical, meaning the center must lie on the line y 6. This gives us k 6.

Substituting k 6 into k 2h gives 6 2h, so h 3. Therefore, the center of the circle is (3, 6).

Distance Calculation

The radius of the circle is the distance from the center (3, 6) to the tangent point (1, 6).

The distance formula gives:

Radius sqrt{(3-1)^2 (6-6)^2} sqrt{4} 2

Equation of the Circle

The standard form of the equation of a circle with center (h, k) and radius r is:

(x - h)^2 (y - k)^2 r^2

Substituting the center (3, 6) and radius 2 gives:

(x - 3)^2 (y - 6)^2 4

Horizontal Line and Normal

The horizontal line passing through the tangent point (1, 6) is y 6. The center of the circle, being the intersection of the line y 6 and the line y 2x, is at (3, 6).

The radius of the circle is the distance from the center (3, 6) to the tangent point (1, 6), which is 2.

Conclusion

The center of the circle is (3, 6), and the radius is 2. The equation of the circle is:

(x - 3)^2 (y - 6)^2 4