JEE Main Mathematics: Section formula in three dimensions Practice Questions
14 free multiple-choice questions on Section formula in three dimensions (Analytical Geometry) for JEE Main Mathematics, each with the correct answer and a step-by-step explanation. Attempt a question, then reveal the solution to check your reasoning.
Practice thousands more JEE Main questions in the app →Q1. The yz-plane divides the line segment joining (−2, 4, 7) and (3, −5, 8) in the ratio
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Correct answer: C — 2 : 3 internally
Let the ratio be k : 1. x-coordinate of the dividing point = (3k − 2)/(k + 1). On the yz-plane x = 0, so 3k − 2 = 0 and k = 2/3. k > 0, so the ratio is 2 : 3 internally.
Q2. Assertion (A): The points (1, 2, 3), (−2, 3, 5) and (7, 0, −2) are collinear. Reason (R): Three points A, B, C are collinear if the vector AB is a scalar multiple of the vector BC.
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Correct answer: D — A is false but R is true
AB = (−3, 1, 2) and BC = (9, −3, −7). 9/−3 = −3, −3/1 = −3 but −7/2 ≠ −3, so AB is not parallel to BC: A is false. The stated condition for collinearity is true, so R is true.
Q3. The xy-plane divides the line joining (1, 2, 3) and (4, 2, 1) in the ratio
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Correct answer: B — 3 : 1 externally
Let the ratio be k : 1; z-coordinate = (k·1 + 3)/(k + 1). Set z = 0: k = −3. A negative k means external division in the ratio 3 : 1 (both points have z > 0, so they lie on the same side).
Q4. The centroid of the triangle with vertices (1, −2, 3), (4, 5, −1) and (−2, 0, 7) is
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Correct answer: B — (1, 1, 3)
Centroid = ((1 + 4 − 2)/3, (−2 + 5 + 0)/3, (3 − 1 + 7)/3). = (1, 1, 3).
Q5. The points A(−2, 3, 5), B(1, 2, 3) and C(7, 0, −1) are collinear. The ratio in which B divides AC is
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Correct answer: C — 1 : 2 internally
AB = (3, −1, −2) and BC = (6, −2, −4) = 2·AB. So AB : BC = 1 : 2 with B between A and C. B divides AC internally in 1 : 2.
Q6. A point R with x-coordinate 4 lies on the line segment joining P(2, 2, 1) and Q(5, 1, −2). The coordinates of R are
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Correct answer: C — (4, 4/3, −1)
R = P + t(Q − P) = (2 + 3t, 2 − t, 1 − 3t). 2 + 3t = 4 gives t = 2/3. R = (4, 2 − 2/3, 1 − 2) = (4, 4/3, −1).
Q7. The line through A(2, 4, 5) and B(3, 5, −4) meets the yz-plane at the point
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Correct answer: A — (0, 2, 23)
Points on AB: (2 + t, 4 + t, 5 − 9t). On the yz-plane x = 0, so t = −2. Point = (0, 2, 5 + 18) = (0, 2, 23); since t < 0 the plane divides AB externally (2 : 3).
Q8. Which of the following is a point of trisection of the line segment joining A(4, −1, −2) and B(−2, 5, 4)?
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Correct answer: A — (2, 1, 0)
Trisection points divide AB in 1 : 2 and 2 : 1. Dividing in 1 : 2: A + (B − A)/3 = (4 − 2, −1 + 2, −2 + 2) = (2, 1, 0). The other is (0, 3, 2); (1, 2, 1) is only the midpoint.
Q9. The point which divides the join of P(1, −2, 3) and Q(3, 0, 5) externally in the ratio 2 : 1 is
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Correct answer: A — (5, 2, 7)
External division in m : n: R = (mQ − nP)/(m − n). R = (2Q − P)/1 = (6 − 1, 0 + 2, 10 − 3) = (5, 2, 7).
Q10. Two vertices of a triangle are (3, −5, 7) and (−1, 7, −6), and its centroid is (1, 1, 1). The third vertex is
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Correct answer: B — (1, 1, 2)
Third vertex = 3G − A − B. x = 3 − 3 + 1 = 1, y = 3 + 5 − 7 = 1, z = 3 − 7 + 6 = 2.
Q11. In triangle ABC with A(0, 0, 0), B(3, 4, 0) and C(0, 0, 10), the bisector of the angle A meets BC at D. Then D is
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Correct answer: C — (2, 8/3, 10/3)
AB = 5 and AC = 10. The internal bisector divides BC in AB : AC = 1 : 2, so BD : DC = 1 : 2. D = (2B + C)/3 = ((6 + 0)/3, 8/3, 10/3) = (2, 8/3, 10/3).
Q12. The mid-point of the segment AB is (2, −1, 4). If A is (5, 3, −2), then B is
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Correct answer: A — (−1, −5, 10)
B = 2M − A. B = (4 − 5, −2 − 3, 8 + 2) = (−1, −5, 10).
Q13. The points on the line through A(1, 2, 3) and B(4, 5, 6) such that AP = 2·PB are
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Correct answer: D — (3, 4, 5) and (7, 8, 9)
AP : PB = 2 : 1, internally or externally. Internal: (A + 2B)/3 = (9/3, 12/3, 15/3) = (3, 4, 5). External: (2B − A)/(2 − 1) = (7, 8, 9).
Q14. The point which divides the line segment joining A(2, −3, 4) and B(8, 0, 10) internally in the ratio 1 : 2 is
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Correct answer: C — (4, −2, 6)
P = (1·B + 2·A)/(1 + 2). x = (8 + 4)/3 = 4, y = (0 − 6)/3 = −2, z = (10 + 8)/3 = 6.
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