Coordinate Geometry

√13 units

Step 1: Use distance formula between A(4, 2) and B(6, 5): AB = √[(6-4)² + (5-2)²]. Step 2: Substitute values: AB = √[2² + 3²]. Step 3: Calculate: AB = √[4 + 9] = √13. Step 4: Since √13 cannot be simplified further, the answer is √13 units. Common mistake: Students might calculate (6-4) + (5-2) = 5, but this is incorrect - we need the square root of sum of squares.

(4, 3)

Step 1: Use section formula for internal division in ratio m:n: P = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n)). Step 2: Here m = 1, n = 2, A(2,1), B(8,7): P = ((1×8 + 2×2)/(1+2), (1×7 + 2×1)/(1+2)). Step 3: Calculate x-coordinate: (8 + 4)/3 = 12/3 = 4. Step 4: Calculate y-coordinate: (7 + 2)/3 = 9/3 = 3. Therefore P(4, 3).

8 units

Step 1: Both points lie on y-axis since x-coordinate is 0 for both. Step 2: For points on same axis, distance = |difference of coordinates|. Step 3: Distance = |5 - (-3)| = |5 + 3| = |8| = 8 units. Step 4: Alternatively using distance formula: d = √[(0-0)² + (5-(-3))²] = √[0 + 64] = 8. When points lie on same axis, we can simply find the absolute difference.

5 or -1

Step 1: Use distance formula: 5 = √[(7-3)² + (2-y)²]. Step 2: Simplify: 5 = √[16 + (2-y)²]. Step 3: Square both sides: 25 = 16 + (2-y)². Step 4: Solve: (2-y)² = 9, so 2-y = ±3. Step 5: Case 1: 2-y = 3 gives y = -1. Case 2: 2-y = -3 gives y = 5. Therefore y = 5 or y = -1.