O-Level Geometry using coordinates

Points (3, 4), (2, k) and (7, 5) are collinear, so gradients are equal. Gradient from (3,4) to (7,5) = (5 - 4)/(7 - 3) = 1/4. Gradient from (3,4) to (2,k) = (k - 4)/(2 - 3) = (k - 4)/(-1). Set equal: (k - 4)/(-1) = 1/4 k - 4 = -1/4 k = 4 - 1/4 = 3.75 = 15/4.

Step 1: Find the gradient of the line through the two points. The gradient formula is m = (y₂ − y₁)/(x₂ − x₁). Take P(−6, 7) as point 1 and R(−1, 1) as point 2: m = (1 − 7)/(−1 − (−6)) Careful with the double negative on the bottom: −1 − (−6) = −1 + 6 = 5 m = (−6)/5 = −6/5 Step 2: Use y = mx + c and one of the points to find the intercept c. Substitute P(−6, 7) and m = −6/5: 7 = (−6/5)(−6) + c (−6/5)(−6) = 36/5 7 = 36/5 + c c = 7 − 36/5 Write 7 as 35/5 so the fractions can be subtracted: c = 35/5 − 36/5 = −1/5 Step 3: Write down the equation. y = (−6/5)x − 1/5 Step 4: Check with the other point R(−1, 1). (−6/5)(−1) − 1/5 = 6/5 − 1/5 = 5/5 = 1 ✓, which is the y-coordinate of R. The equation of PR is y = −6/5 x − 1/5 (equivalently 6x + 5y + 1 = 0).

Step 1: Write the distance between the two points and square both sides. The distance between B(1, 0) and A(k, 2) is AB = √((k − 1)² + (2 − 0)²) We are told this distance equals √(2k + 2). Squaring both sides removes both square roots at once: (k − 1)² + (2 − 0)² = 2k + 2 Step 2: Expand the bracket and collect everything on one side. (k − 1)² = k² − 2k + 1, and (2 − 0)² = 4, so k² − 2k + 1 + 4 = 2k + 2 k² − 2k + 5 = 2k + 2 Subtract 2k and 2 from both sides: k² − 2k − 2k + 5 − 2 = 0 k² − 4k + 3 = 0 Step 3: Factorise the quadratic. We need two numbers that multiply to +3 and add to −4. Those numbers are −1 and −3. k² − 4k + 3 = (k − 1)(k − 3) = 0 A product is zero only when a factor is zero, so k − 1 = 0 giving k = 1, or k − 3 = 0 giving k = 3 Step 4: Check both values are usable, since √(2k + 2) must be a real, non-negative length. k = 1: 2k + 2 = 4, so the stated distance is √4 = 2; and (1 − 1)² + 2² = 0 + 4 = 4 ✓ k = 3: 2k + 2 = 8, so the stated distance is √8; and (3 − 1)² + 2² = 4 + 4 = 8 ✓ Both values work. k = 1 or k = 3.