Question

Let a circle \( C \) with radius \( r \) passes through four distinct points \( (0, 0), (k, 3k), (2, 3), (-1, 5) \), such that \( k \neq 0 \), then \( (10k + 2r^2) \) is equal to:

• A. 35
• B. 34
• C. 27
• D. 32

Hint:

For a circle passing through given points, use the general form of the equation of the circle and substitute the points to find the values of \( g \) and \( f \).

Answer 1

The Correct Option is B

The general equation of a circle is given by \( x^2 + y^2 + 2gx + 2fy + c = 0 \). Given the points \( (0, 0), (k, 3k), (2, 3), (-1, 5) \) lie on the circle, we substitute them into the general equation. 1. Substituting \( (0, 0) \): \[ 0^2 + 0^2 + 2g(0) + 2f(0) + c = 0 \quad \Rightarrow \quad c = 0 \] The equation simplifies to \( x^2 + y^2 + 2gx + 2fy = 0 \). 2. Substituting \( (k, 3k) \): \[ k^2 + (3k)^2 + 2gk + 2f(3k) = 0 \] \[ k^2 + 9k^2 + 2gk + 6fk = 0 \quad \Rightarrow \quad 10k^2 + 2k(g + 3f) = 0 \] This implies \( g + 3f = -5k \). 3. Substituting \( (2, 3) \): \[ 2^2 + 3^2 + 2g(2) + 2f(3) = 0 \] \[ 4 + 9 + 4g + 6f = 0 \quad \Rightarrow \quad 4g + 6f = -13 \] 4. Substituting \( (-1, 5) \): \[ (-1)^2 + 5^2 + 2g(-1) + 2f(5) = 0 \] \[ 1 + 25 - 2g + 10f = 0 \quad \Rightarrow \quad -2g + 10f = -26 \quad \Rightarrow \quad g - 5f = 13 \] Solving the system of equations \( g + 3f = -5k \), \( 4g + 6f = -13 \), and \( g - 5f = 13 \) allows us to find the values of \( g \), \( f \), and \( k \). Subsequently, we can determine \( r^2 \) and then compute \( 10k + 2r^2 \). The computed value of \( 10k + 2r^2 \) is 34.