Question

If the distance between the points $(-1, -3)$ and $(x, 9)$ is 13 units, then find the values of $x$.

Hint:

Always square both sides while using the distance formula to eliminate the square root safely.

Answer 1

Step 1: Use distance formula.
\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = 13 \] \[ \sqrt{(x + 1)^2 + (9 + 3)^2} = 13 \Rightarrow \sqrt{(x + 1)^2 + 144} = 13 \]
Step 2: Simplify.
\[ (x + 1)^2 + 144 = 169 \Rightarrow (x + 1)^2 = 25 \Rightarrow x + 1 = \pm 5 \] \[ x = 4 \text{ or } x = -6 \] Wait — to recheck calculation: $(x+1)^2=25 \Rightarrow x=4$ or $x=-6$.

Step 3: Conclusion.
Hence, $x = 4$ or $x = -6$.