Key Concepts and Formulas

• Standard Form of a Circle: The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$. Its center is $(-g, -f)$ and its radius is $r = \sqrt{g^2 + f^2 - c}$.
• Chord Property: A perpendicular drawn from the center of a circle to a chord bisects the chord.
• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Pythagorean Theorem: In a right-angled triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse and $a$ and $b$ are the other two sides.

Step-by-Step Solution

Step 1: Determine the properties of the first circle.

The equation of the first circle is given as $x^2 + y^2 - 2x - 6y + 6 = 0$. We want to find its center and radius.

Comparing this with the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, we have:

• $2g = -2 \implies g = -1$
• $2f = -6 \implies f = -3$
• $c = 6$

The center of this circle, let's call it $O_1$, is $(-g, -f) = (-(-1), -(-3)) = (1, 3)$. The radius of this circle, let's call it $r_1$, is $\sqrt{g^2 + f^2 - c} = \sqrt{(-1)^2 + (-3)^2 - 6} = \sqrt{1 + 9 - 6} = \sqrt{4} = 2$.

Explanation: We identified $g, f,$ and $c$ by comparing the given equation with the standard form. Using these values, we calculated the center and radius using the respective formulas.

Step 2: Identify the properties of the second circle.

Let the second circle be 'C'. Its center is given as $O_2(2, 1)$. Let its radius be $R$. This is what we need to find.

Explanation: We are given the center of the second circle and we need to determine its radius.

Step 3: Apply the given condition to establish a geometric relationship.

The problem states that "one of the diameters of the circle $x^2 + y^2 - 2x - 6y + 6 = 0$ is a chord of another circle 'C'". Let the diameter of the first circle be $AB$. The length of this diameter is $2r_1 = 2(2) = 4$. Since $AB$ is a diameter of the first circle, its midpoint is the center $O_1(1, 3)$. Now, $AB$ is also a chord of the second circle 'C'.

Explanation: The diameter of the first circle is a chord of the second circle. The midpoint of this chord is the center of the first circle.

Step 4: Form a right-angled triangle using the chord property.

Consider the second circle 'C' with center $O_2(2, 1)$ and radius $R$. The line segment $AB$ is a chord of 'C'. The perpendicular from the center $O_2$ to the chord $AB$ bisects $AB$. Since $O_1$ is the midpoint of $AB$, the line segment connecting $O_2$ to $O_1$ is perpendicular to the chord $AB$. We can form a right-angled triangle with vertices $O_2$, $O_1$, and one endpoint of the chord, say $A$. The hypotenuse of this triangle is $O_2A$, which is the radius $R$ of circle 'C'. One leg is $O_1A$, which is half the length of the chord $AB$. Since $AB$ is the diameter of the first circle, $O_1A = r_1 = 2$. The other leg is $O_2O_1$, which is the distance between the centers of the two circles.

Explanation: We are using the property that a line from the center of a circle to the midpoint of a chord is perpendicular to the chord. This allows us to construct a right triangle.

Step 5: Calculate the distance between the centers $O_1$ and $O_2$.

Using the distance formula for $O_1(1, 3)$ and $O_2(2, 1)$: $O_1O_2 = \sqrt{(2 - 1)^2 + (1 - 3)^2} = \sqrt{(1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}$

Explanation: We use the distance formula to find the distance between the centers of the two circles.

Step 6: Apply the Pythagorean theorem to find the radius $R$.

From the right-angled triangle formed by $O_2$, $O_1$, and $A$: $R^2 = (O_1A)^2 + (O_1O_2)^2$ We know $O_1A = r_1 = 2$ and $O_1O_2 = \sqrt{5}$. $R^2 = (2)^2 + (\sqrt{5})^2 = 4 + 5 = 9$ $R = \sqrt{9} = 3$

Explanation: We apply the Pythagorean theorem to relate the radius of the second circle to the distance between the centers and the radius of the first circle.

Common Mistakes & Tips:

• Diagram: Draw a diagram to visualize the problem.
• Chord Property: Remember that the line joining the center to the midpoint of the chord is perpendicular to the chord.
• Pythagorean Theorem: Be careful while applying the Pythagorean theorem and identify the correct sides of the right-angled triangle.

Summary

This problem combines the algebraic representation of circles with geometric properties. We first found the center and radius of the first circle. Then, using the fact that a diameter of the first circle is a chord of the second, we constructed a right-angled triangle to apply the Pythagorean theorem and find the radius of the second circle.

The final answer is 3.

Final Answer

The final answer is \boxed{3}.