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

Given the points on the circle: \( (0, 0), (k, 3k), (2, 3), (-1, 5) \), the general equation of a circle is: \[ x^2 + y^2 + 2gx + 2fy + c = 0 \] Substitute the given points into the equation: 1. Substituting \( (0, 0) \) into the equation gives: \[ 0^2 + 0^2 + 2g(0) + 2f(0) + c = 0 \quad \Rightarrow \quad c = 0 \] Thus, the equation becomes: \[ x^2 + y^2 + 2gx + 2fy = 0 \] 2. Substituting \( (k, 3k) \) into the equation: \[ k^2 + (3k)^2 + 2gk + 2f(3k) = 0 \] \[ k^2 + 9k^2 + 2gk + 6fk = 0 \quad \Rightarrow \quad 10k^2 + 2k(g + 3f) = 0 \] Thus: \[ g + 3f = -5k \] 3. Substituting \( (2, 3) \) into the equation: \[ 2^2 + 3^2 + 2g(2) + 2f(3) = 0 \] \[ 4 + 9 + 4g + 6f = 0 \quad \Rightarrow \quad 4g + 6f = -13 \] Now solving these equations will give you the values of \( g \) and \( f \), and you can find \( r^2 \), the radius squared, to finally calculate \( 10k + 2r^2 \). After solving the system, we find that the value of \( 10k + 2r^2 \) is 34.