Using the Distance Formula to Find an Unknown

When a question fixes the length between two points and hides a coordinate, square the distance formula to get a quadratic in the unknown. Squaring is what introduces the two answers students so often miss.

The points $A(-1, 4)$ and $B(3, k)$ are such that the length $AB = 5$. Find the possible values of $k$. [4]

The working

Step 1, write $AB^2$ using the distance formula (square both sides to avoid surds): $AB^2 = (3 - (-1))^2 + (k - 4)^2 = 5^2 \quad \text{(M1)}$

Step 2, simplify the known part: $(4)^2 + (k - 4)^2 = 25 \;\Rightarrow\; 16 + (k - 4)^2 = 25 \quad \text{(A1)}$

Step 3, solve for $k$: $(k - 4)^2 = 9 \;\Rightarrow\; k - 4 = \pm 3 \quad \text{(M1)}$ $k = 4 + 3 = 7 \qquad \text{or} \qquad k = 4 - 3 = 1 \quad \text{(A1)}$

Where the marks are won and lost

• Squaring the distance ($AB^2 = 25$) avoids carrying a square root through the algebra, cleaner and less error-prone.
• $(k - 4)^2 = 9$ gives $k - 4 = \pm 3$, hence two values. Taking only the positive root loses $k = 1$.
• $(3 - (-1))^2 = 4^2 = 16$: mind the double negative in the $x$-difference.

Common mistakes

• Writing $3 - (-1) = 2$ (sign error), giving the wrong constant.
• Forgetting the $\pm$ and reporting only $k = 7$.
• Leaving the answer as $(k-4)^2 = 9$ without solving.

Full method: Gradient, Midpoint & Length notes. Topic home: Straight-Line Graphs pillar.