Determining the Radius and Center of a Circle Tangent to a Line

We are given a circle with a center on the line (y 2x) and a tangent line (x 1) at the point ((1, 6)). We aim to determine the coordinates of the circle's center and its radius. Let's walk through the solution step by step.

Step 1: Determine the Center of the Circle

Let the coordinates of the center of the circle be ((h, k)). Since the center lies on the line (y 2x), we have:

Given (k 2h), we set:

(k 2h quad …..1)

This tells us that the coordinates of the center are ((h, 2h)).

Step 2: Determine the Radius of the Circle

The line (x 1) is tangent to the circle at the point ((1, 6)). The radius at the point of tangency is perpendicular to the tangent line. The slope of the tangent line (x 1) is undefined (it is a vertical line), so the radius must be horizontal. The slope of a horizontal line is 0, and the equation of the normal (which passes through ((1, 6))) is (y 6).

The center of the circle lies on the line (y 2x) and also on the line (y 6). We can thus solve for (h) and (k):

Since (k 6) and (k 2h), we get:

(6 2h)

Hence, (h 3).

Therefore, the center of the circle is ((3, 6)).

Step 3: Calculate the Radius

The radius is the perpendicular distance from the center of the circle ((3, 6)) to the point of tangency ((1, 6)). This distance is simply the horizontal distance, which is:

(r 3 - 1 2)

Step 4: Write the Equation of the Circle

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

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

Therefore, the equation of the circle is:

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

Final Answer

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

Key Points:

The center of the circle lies on the line (y 2x). The radius is the perpendicular distance from the center to the point of tangency. The equation of a circle with center ((h, k)) and radius (r) is ((x - h^2) (y - k^2) r^2).