Class CBSE Class 12 Mathematics Three Dimensional Geometry Q #1427
KNOWLEDGE BASED
REMEMBER
2 Marks 2025 AISSCE(Board Exam) VSA
A man needs to hang two lanterns on a straight wire whose end points have coordinates $A(4,1,-2)$ and $B(6,2,-3)$. Find the coordinates of the points where he hangs the lanterns such that these points trisect the wire AB.
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Detailed Solution

Step 1: Understanding Trisection

Trisection means dividing the line segment into three equal parts. Let the points that trisect the wire AB be P and Q. Then AP = PQ = QB.

Step 2: Finding the coordinates of point P

Since P divides AB in the ratio 1:2, we can use the section formula to find the coordinates of P. The section formula for a point dividing a line segment joining points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in the ratio m:n is given by: $P(x, y, z) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right)$ Here, $A(4, 1, -2)$, $B(6, 2, -3)$, $m = 1$, and $n = 2$. $P(x, y, z) = \left(\frac{1(6) + 2(4)}{1+2}, \frac{1(2) + 2(1)}{1+2}, \frac{1(-3) + 2(-2)}{1+2}\right)$ $P(x, y, z) = \left(\frac{6 + 8}{3}, \frac{2 + 2}{3}, \frac{-3 - 4}{3}\right)$ $P(x, y, z) = \left(\frac{14}{3}, \frac{4}{3}, \frac{-7}{3}\right)$

Step 3: Finding the coordinates of point Q

Since Q divides AB in the ratio 2:1, we can use the section formula to find the coordinates of Q. Here, $A(4, 1, -2)$, $B(6, 2, -3)$, $m = 2$, and $n = 1$. $Q(x, y, z) = \left(\frac{2(6) + 1(4)}{2+1}, \frac{2(2) + 1(1)}{2+1}, \frac{2(-3) + 1(-2)}{2+1}\right)$ $Q(x, y, z) = \left(\frac{12 + 4}{3}, \frac{4 + 1}{3}, \frac{-6 - 2}{3}\right)$ $Q(x, y, z) = \left(\frac{16}{3}, \frac{5}{3}, \frac{-8}{3}\right)$

Final Answer: The coordinates of the points are $P\left(\frac{14}{3}, \frac{4}{3}, \frac{-7}{3}\right)$ and $Q\left(\frac{16}{3}, \frac{5}{3}, \frac{-8}{3}\right)$.

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Pedagogical Audit
Bloom's Analysis: This is an REMEMBER question because the student needs to recall the section formula and apply it to find the coordinates of the points.
Knowledge Dimension: PROCEDURAL
Justification: The question requires the student to apply a specific procedure (section formula) to calculate the coordinates of the points that trisect the line segment.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding and application of the section formula, a standard concept in 3D geometry as per the textbook.

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