Find the values of y for which the distance between the points P(2, -3) and Q(10, y) is 10 units.
y = 3 or y = -9
Explanation
This question tests the application of the distance formula derived from the Pythagoras theorem. The context shows how to find distance between two points using the formula PQ = √[(x₂ - x₁)² + (y₂ - y₁)²]. Students need to substitute the given coordinates, set up the equation with distance = 10, and solve the resulting quadratic equation. The key insight is recognizing that squaring both sides eliminates the square root, and taking square root of 36 gives two possible values (±6), leading to two solutions for y.
Solution Steps
Step 1: Apply the distance formula: PQ = √[(x₂ - x₁)² + (y₂ - y₁)²]
Step 2: Substitute P(2, -3) and Q(10, y): 10 = √[(10 - 2)² + (y - (-3))²]
Step 3: Simplify: 10 = √[8² + (y + 3)²] = √[64 + (y + 3)²]
Step 4: Square both sides: 100 = 64 + (y + 3)²
Step 5: Solve: (y + 3)² = 36, so y + 3 = ±6
Step 6: Therefore, y = 3 or y = -9