Coordinate Geometry

University:
Stanford University
Course:
Introduction to Mathematical Thinking
Academic year:

2023
Views:

405

Pages:

3
Author:

MAlphen

Coordinate Geometry 1 (a) A straight-line l has equation 3𝑦 − 2𝑥 = 5. Find the equation of a line parallel to l and passes through the point (−1, 4). (b) The distance between two points 𝐴 (𝑘, 2) and 𝐵 (1, 0) is √2𝑘 + 2 units. Find the values of k. [2] [2] 2 The diagram shows triangle PQR with coordinates 𝑃(−8, 0), 𝑄(10, 0) and 𝑅(−5, 9).The line QR cuts the y-axis at the point S. Find (a) the length of QR, (b) the equation of the line QR, (c) the coordinates of S, (d) the area of quadrilateral OPRS. [2] [2] [2] [2] 3 The figure shows a triangle ABC with 𝐵 (−2, 4), 𝐶 (2, 2) and 𝐴 (𝑎, 𝑏) . The gradients of BC, AC and AB are −2𝑛, 2𝑛, n respectively. (a) Find (i) the value of n, (ii) the coordinates of A, (iii) the length of BC. (b) Given that the area of triangle ABC is 12 units2, find the length of the perpendicular line drawn from A to BC. [2] [2] [2] [2] 1 Solutions: 2 (a) 𝑦 = 𝑥 + 𝑐 (b) √(𝑘 − 1)2 + (2 − 0)2 = √2𝑘 + 2 3 (4) = 2 (−1) + 𝐶 𝑘2 − 2𝑘 + 1 + 4 = 2𝑘 + 2 𝑐= 3 2 14 ∴𝑦= 𝑥+ 𝑘 − 4𝑘 + 3 = 0 (𝑘 − 1)(𝑘 − 3) = 0 3 14 2 3 𝑘 = 1 or 𝑘 = 3 3 2 14 3 3 Ans: (a) 𝑦 = 𝑥 + (b) 𝑘 = 1 or 𝑘 = 3 2 Solutions: (a) 𝑄𝑅 = √(10 + 5)2 + (0 − 9)2 = √306 = 17.4928556845 = 17.5 units (b) 𝑚 = = 0−9 10−(−5) −9 15 3 = −5 3 0 = − 5 (10) + 𝑐 𝑐=6 3 ⇒𝑦 =− 𝑥+6 5 1 (d) 1 (3)(9) + (9 + 6)(5) 2 = 27 2 + 75 2 ∆𝑅𝑃𝑄 − ∆𝑆𝑂𝑄 2 = 51 units2 OR = 81 − 30 = 51 units2 3 (d) Ans: (a) 𝑄𝑅 = 17.5 units (b) 𝑦 = − 𝑥 + 6 (c) 𝑆(0, 6) (d) 51 units2 5 3 Solutions: (a)(i) Gradient of BC 4−2 = (iii) Length of BC = √(−2 − 2)2 + (4 − 2)2 −2−2 1 =− −2𝜋 = − 1 = √20 = 4.4721 = 4.47 units 2 1 2 𝑛= 4 (ii) Gradient of 𝐴𝐶 = 2𝜋 𝑏−2 1 = 2 (4) 2𝑏 − 4 = 𝑎 − 2 2𝑏 − 𝑎 = 2------- (1) Gradient of 𝐴𝐵 = 𝑛 𝑏−4 1 =4 𝑎+2 4𝑏 − 16 = 𝑎 + 2 4𝑏 − 18 = 𝑎 --------- (2) 𝑎−2 1 (b) × √20 × ℎ = 12 2 ℎ = 5.3665 ℎ = 5.37 units Sub (2) into (1) 2𝑏 − (4𝑏 − 18) = 2 −2𝑏 + 18 = 2 𝑏=8 𝑎 = 4(8) − 18 𝑎 = 14 𝐴(14,8) 1 Ans: (a)(i) 𝑛 = (ii) 𝐴(14, 8) (iii) Length of BC = 4.47units (b) ℎ = 5.37𝑢𝑛𝑖𝑡𝑠 4