Key Concepts and Formulas

• Equation of a Circle: The standard equation of a circle with center $(h, k)$ and radius $R$ is given by $(x - h)^2 + (y - k)^2 = R^2$.
• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Circle's Center Property: The center of a circle is equidistant from all points on its circumference. If a circle passes through points $P_1$ and $P_2$, and its center is $C$, then $CP_1 = CP_2 = R$.

Step-by-Step Solution

1. Define the Center and Use the Line Equation

Let the center of the circle be $C(h, k)$. The center lies on the line $y - 4x + 3 = 0$. Why this step? This gives us a relationship between $h$ and $k$, which we'll use to solve for the center's coordinates.

Substituting $(h, k)$ into the line equation: $k - 4h + 3 = 0$ $k = 4h - 3 \quad \text{(Equation 1)}$

2. Apply the Equidistance Property

The circle passes through $P_1(2, 3)$ and $P_2(4, 5)$. The distance from the center $C(h, k)$ to $P_1$ equals the distance from $C(h, k)$ to $P_2$. Why this step? This is a fundamental property of circles allowing us to create an equation to solve for the center.

Using the distance formula: $CP_1 = \sqrt{(h - 2)^2 + (k - 3)^2}$ $CP_2 = \sqrt{(h - 4)^2 + (k - 5)^2}$

Equating the distances: $\sqrt{(h - 2)^2 + (k - 3)^2} = \sqrt{(h - 4)^2 + (k - 5)^2}$

Squaring both sides to eliminate the square roots: $(h - 2)^2 + (k - 3)^2 = (h - 4)^2 + (k - 5)^2$

3. Expand and Simplify the Equation

Expand the squared terms: $(h^2 - 4h + 4) + (k^2 - 6k + 9) = (h^2 - 8h + 16) + (k^2 - 10k + 25)$ Why this step? Expanding allows us to combine like terms and simplify the equation. The $h^2$ and $k^2$ terms will cancel out.

Combine terms: $h^2 + k^2 - 4h - 6k + 13 = h^2 + k^2 - 8h - 10k + 41$

Subtract $h^2 + k^2$ from both sides: $-4h - 6k + 13 = -8h - 10k + 41$

Rearrange terms: $4h + 4k = 28$

Divide by 4: $h + k = 7 \quad \text{(Equation 2)}$

4. Solve the System of Equations

We have the system:

1. $k = 4h - 3$
2. $h + k = 7$

Why this step? We have two equations and two unknowns, enabling us to solve for $h$ and $k$.

Substitute Equation 1 into Equation 2: $h + (4h - 3) = 7$ $5h - 3 = 7$ $5h = 10$ $h = 2$

Substitute $h = 2$ back into Equation 1: $k = 4(2) - 3$ $k = 8 - 3$ $k = 5$

The center of the circle is $C(2, 5)$.

5. Calculate the Radius

Use the distance formula between the center $C(2, 5)$ and $P_1(2, 3)$: Why this step? The radius is the distance from the center to any point on the circle.

$R = \sqrt{(2 - 2)^2 + (5 - 3)^2}$ $R = \sqrt{0^2 + 2^2}$ $R = \sqrt{4}$ $R = 2$

6. Verify the Radius

Check using point $P_2(4, 5)$: $R = \sqrt{(2 - 4)^2 + (5 - 5)^2}$ $R = \sqrt{(-2)^2 + 0^2}$ $R = \sqrt{4}$ $R = 2$

The radius of the circle is 2.

Common Mistakes & Tips

• Algebraic Errors: Double-check expanding squared terms and combining like terms to avoid sign errors.
• Simplification: Simplify equations whenever possible (e.g., dividing by 4 in $4h + 4k = 28$).
• Equation Substitution: Ensure correct substitution when solving the system of equations.

Summary

We found the equation of a circle given two points it passes through and a condition on its center. We translated the geometric conditions into algebraic equations. By using the property that the center is equidistant from points on the circumference, we set up a system of linear equations to find the coordinates of the center. Then we calculated the radius using the distance formula.

The final answer is 2. This corresponds to option (A).

The final answer is $\boxed{2}$, which corresponds to option (A).