NBT Mathematics: Analytical Geometry — Questions and Answers
Question 1: Calculate the distance between points A(−1, 3) and B(5, −5).
- 8
- 10 (Correct answer)
- √20
- 14
Correct answer: 10
Using the distance formula: d = √[(5−(−1))² + (−5−3)²] = √[6² + (−8)²] = √[36 + 64] = √100 = 10.
Question 2: M is the midpoint of P(−4, 6) and Q(2, −2). What are the coordinates of M?
- (−1, 2) (Correct answer)
- (1, −2)
- (−2, 4)
- (3, −4)
Correct answer: (−1, 2)
Midpoint M = ((x₁+x₂)/2, (y₁+y₂)/2) = ((−4+2)/2, (6+(−2))/2) = (−2/2, 4/2) = (−1, 2).
Question 3: A line has a gradient of 3/4. What is the gradient of a line perpendicular to it?
- 3/4
- 4/3
- −4/3 (Correct answer)
- −3/4
Correct answer: −4/3
For perpendicular lines, m₁ × m₂ = −1. So m₂ = −1 ÷ (3/4) = −4/3. The perpendicular gradient is the negative reciprocal of the original.
Question 4: A circle has centre (3, −2) and radius 5. Which equation represents this circle?
- (x + 3)² + (y − 2)² = 25
- (x − 3)² + (y + 2)² = 25 (Correct answer)
- (x − 3)² + (y − 2)² = 5
- (x + 3)² + (y + 2)² = 25
Correct answer: (x − 3)² + (y + 2)² = 25
The standard form is (x − h)² + (y − k)² = r², where (h, k) is the centre and r is the radius. With centre (3, −2) and r = 5: (x − 3)² + (y − (−2))² = 25 → (x − 3)² + (y + 2)² = 25.
Question 5: Are the points A(0, 0), B(2, 3) and C(4, 7) collinear?
- Yes, because the gradients AB and BC are equal
- No, because the distance AC does not equal AB + BC
- No, because the gradients AB and BC are different (Correct answer)
- Yes, because all three points lie on the same x-axis
Correct answer: No, because the gradients AB and BC are different
Gradient AB = (3−0)/(2−0) = 3/2. Gradient BC = (7−3)/(4−2) = 4/2 = 2. Since 3/2 ≠ 2, the gradients differ, so the three points are NOT collinear — they do not lie on the same straight line.
Question 6: Find the equation of the straight line passing through (1, 4) and (3, −2).
- y = 3x + 1
- y = −3x + 7 (Correct answer)
- y = −3x + 1
- y = 3x + 7
Correct answer: y = −3x + 7
Gradient m = (−2 − 4)/(3 − 1) = −6/2 = −3. Using point (1, 4): y − 4 = −3(x − 1) → y = −3x + 3 + 4 → y = −3x + 7.
NBT Mathematics: Analytical Geometry
Calculate the distance between points A(−1, 3) and B(5, −5).