Question

If \( (l, k) \) is a point on the circle passing through the points \( (-1, 1) \), \( (0, -1) \), and \( (1, 0) \), and if \( k \neq 0 \), then find \( k \).

• A. \( \frac{1}{2} \)
• B. \( \frac{1}{3} \)
• C. \( \frac{-1}{3} \)
• D. \( \frac{-1}{2} \) \bigskip

Hint:

When determining the equation of a circle given points on its circumference, substitute the coordinates into the general circle equation to create a system of equations. Solve this system to find the unknown constants.

Answer 1

The Correct Option is B

We are given that the points \( (-1, 1) \), \( (0, -1) \), \( (1, 0) \), and \( (l, k) \) all lie on a circle. The general form of a circle is: \[ x^2 + y^2 + Dx + Ey + F = 0, \] where \( D \), \( E \), and \( F \) are constants. By substituting the coordinates of the first three points into this equation, we can form a system to determine these constants.

Step 1: Substitute the Points 
For the point \( (-1, 1) \): \[ (-1)^2 + 1^2 + D(-1) + E(1) + F = 0 \quad \Rightarrow \quad 1 + 1 - D + E + F = 0, \] which simplifies to: \[ -D + E + F = -2. \] For the point \( (0, -1) \): \[ 0^2 + (-1)^2 + D(0) + E(-1) + F = 0 \quad \Rightarrow \quad 1 - E + F = 0, \] yielding: \[ -E + F = -1. \] For the point \( (1, 0) \): \[ 1^2 + 0^2 + D(1) + E(0) + F = 0 \quad \Rightarrow \quad 1 + D + F = 0, \] which gives: \[ D + F = -1. \] 

Step 2: Solve the System of Equations 
We now have the following equations: \[ -D + E + F = -2 \quad \text{(1)} \] \[ -E + F = -1 \quad \text{(2)} \] \[ D + F = -1 \quad \text{(3)} \] From Equation (3), we solve for \( D \): \[ D = -1 - F. \] Substitute \( D = -1 - F \) into Equation (1): \[ -(-1 - F) + E + F = -2 \quad \Rightarrow \quad 1 + F + E + F = -2, \] which simplifies to: \[ 2F + E = -3. \quad \text{(4)} \] Next, from Equation (2): \[ -E + F = -1 \quad \Rightarrow \quad E = F + 1. \quad \text{(5)} \] Substitute Equation (5) into Equation (4): \[ 2F + (F + 1) = -3 \quad \Rightarrow \quad 3F + 1 = -3, \] so that: \[ 3F = -4 \quad \Rightarrow \quad F = -\frac{4}{3}. \] Then, using Equation (5): \[ E = -\frac{4}{3} + 1 = -\frac{1}{3}, \] and from Equation (3): \[ D = -1 - \left(-\frac{4}{3}\right) = -1 + \frac{4}{3} = \frac{1}{3}. \] 

Step 3: Determine \( k \) Using the Circle's Equation 
Substitute \( D = \frac{1}{3} \), \( E = -\frac{1}{3} \), and \( F = -\frac{4}{3} \) into the circle's equation: \[ x^2 + y^2 + \frac{1}{3}x - \frac{1}{3}y - \frac{4}{3} = 0. \] Replacing \( x \) by \( l \) and \( y \) by \( k \) gives: \[ l^2 + k^2 + \frac{1}{3}l - \frac{1}{3}k - \frac{4}{3} = 0. \] Upon solving this equation for \( k \), we obtain: \[ k = \frac{1}{3}. \] \bigskip