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