10 questions · Form 4 Additional Mathematics Bab 7: Coordinate Geometry
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Prefer reading to quizzing? All 10 questions are listed below with the answer and explanation under each one.
1. The points A(1, 3), B(4, k), and C(7, 11) are collinear. Find the value of k.
Answer: A
Gradient AB = Gradient AC => k - 34 - 1 = 11 - 37 - 1 => k - 33 = 86 = 43 => k - 3 = 4 => k = 7.
2. If line L₁ has equation 2x - 3y + 6 = 0 and line L₂ is perpendicular to L₁, find the gradient of L₂.
Answer: A
Rearranging L₁: 3y = 2x + 6 => y = (23)x + 2. So gradient m₁ = 23. For L₂ ⊥ L₁: m₂ = -1m₁ = -1 / (23) = -32.
3. Find the perpendicular distance from the origin (0,0) to the midpoint of the line joining A(2, 6) and B(6, 2).
Answer: A
Midpoint M = (2+62, 6+22) = (4, 4). Distance OM = √4² + 4² = √32 = 4√2 units.
4. The lines y = 2x + 1 and y = kx - 4 are perpendicular. Find the value of k.
Answer: A
m₁ = 2, m₂ = k. Since they are perpendicular, m₁ m₂ = -1 => 2k = -1 => k = -12.
5. The point P(k, 5) divides the line segment joining A(-1, 2) and B(7, 8) internally in the ratio m : n. Find the ratio m : n.
Answer: A
Using y-coordinate: y = n y₁ + m y₂m + n => 5 = 2n + 8mm + n => 5m + 5n = 8m + 2n => 3n = 3m => mn = 11 => 1 : 1.
6. Find the y-intercept of the line that is perpendicular to 4x - 2y + 5 = 0 and passes through (2, 1).
Answer: A
4x - 2y + 5 = 0 => 2y = 4x + 5 => y = 2x + 2.5, so m₁ = 2. Perpendicular gradient m₂ = -12. Line: y - 1 = -12(x - 2) => y = -12 x + 1 + 1 => y = -12 x + 2. The y-intercept is 2.
7. If the area of a triangle with vertices (0, 0), (k, 2), and (3, 4) is 7 square units, find the positive value of k.
Answer: A
Area = 12 | 0(2) + k(4) + 3(0) - (0(k) + 2(3) + 4(0)) | = 12 | 4k - 6 | = 7 => |4k - 6| = 14 => 4k - 6 = 14 => 4k = 20 => k = 5.
8. Find the equation of the line passing through (4, -1) and parallel to the line 3x + y = 7.
Answer: A
3x + y = 7 => y = -3x + 7, so m = -3. Equation: y - (-1) = -3(x - 4) => y + 1 = -3x + 12 => y = -3x + 11.
9. Point P divides AB internally in the ratio 3 : 1. If A is (2, 1) and P is (8, 7), find the coordinates of B.
Answer: A
Let B = (x, y). P = (1(2) + 3x4, 1(1) + 3y4) = (8, 7). x-coord: 2 + 3x4 = 8 => 2 + 3x = 32 => 3x = 30 => x = 10. y-coord: 1 + 3y4 = 7 => 1 + 3y = 28 => 3y = 27 => y = 9. Thus B = (10, 9).
10. Line L passes through (2, 5) and (4, 9). Line N passes through (1, 1) and (3, k). If L is parallel to N, find k.
Answer: A
Gradient of L: m_L = 9 - 54 - 2 = 42 = 2. Since L ∥ N, m_N = 2 => k - 13 - 1 = 2 => k - 12 = 2 => k - 1 = 4 => k = 5.