Question

Find the values of \( y \) for which the distance between the points (5, -3) and (13, y) is 10 units.

Hint:

When using the distance formula, always square both sides to eliminate the square root and solve for the variable.

Answer 1

Step 1: Recall the distance formula.
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 2: Substitute the given values.
\[ 10 = \sqrt{(13 - 5)^2 + (y - (-3))^2} \] \[ 10 = \sqrt{8^2 + (y + 3)^2} \]
Step 3: Simplify.
\[ 10 = \sqrt{64 + (y + 3)^2} \] \[ 100 = 64 + (y + 3)^2 \] \[ (y + 3)^2 = 36 \]
Step 4: Solve for \( y \).
\[ y + 3 = \pm 6 \] Case 1: \( y + 3 = 6 \Rightarrow y = 3 \)
Case 2: \( y + 3 = -6 \Rightarrow y = -9 \)
Step 5: Conclusion.
Hence, the values of \( y \) are \( \boxed{3 \text{ and } -9} \).