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Introduction to Three Dimensional Geometry - Previous Year Questions

⏱️ Est. Revision Time: 15 Mins
⚡ Key Formulas at a Glance
Distance Formula 3D
d = √[(x2−x1)²+(y2−y1)²+(z2−z1)²]
Section Formula 3D
P = ((mx2+nx1)/(m+n), ...)
Distance from XY-plane
|z|
🔍
📅 Year 2024
Q1
CBSE Board Exam 20243 Marks
Find the coordinates of the point which divides the line joining A(2, −3, 4) and B(8, 0, 10) in the ratio 2 : 1 externally.
▶ Show Detailed Solution
Section formula (external, m : n = 2 : 1): P = (mx₂ − nx₁)/(m − n), etc.
x = (2×8 − 1×2)/(2−1) = 14; y = (2×0 − 1×(−3))/1 = 3; z = (2×10 − 1×4)/1 = 16.
P = (14, 3, 16).
Q2
CBSE Board Exam 20242 Marks
A point is at a distance of 6 units from the origin. If its direction cosines are proportional to 2, −1, 2, find the coordinates of the point.
▶ Show Detailed Solution
Direction ratios 2, −1, 2 ⇒ |r| = √(4+1+4) = 3.
l = 2/3, m = −1/3, n = 2/3.
Point = (6 × 2/3, 6 × (−1/3), 6 × 2/3) = (4, −2, 4).
(4, −2, 4).
📅 Year 2023
Q1
CBSE Board Exam 20233 Marks
Show that the points A(1, 2, 3), B(4, 0, 4) and C(−2, 4, 2) are collinear.
▶ Show Detailed Solution
AB = (3, −2, 1), AC = (−3, 2, −1).
AC = −AB, so the vectors are parallel and A is common ⇒ collinear.
Proved: points are collinear.