CBSE 2025 · Region 7 · Set 1 · Q35 · 5 marks
Find the point Q on the line $\displaystyle \frac{2 \mathrm{x}+4}{6}=\frac{\mathrm{y}+1}{2}=\frac{-2 \mathrm{z}+6}{-4}$ at a distance of $\displaystyle 3 \sqrt{2}$ from the point $\displaystyle \mathrm{P}(1,2,3)$.Find the image of the point $\displaystyle (-1,5,2)$ in the line $\displaystyle \frac{2 \mathrm{x}-4}{2}=\frac{\mathrm{y}}{2}=\frac{2-\mathrm{z}}{3}$. Find the length of the line segment joining the points (given point and the image point).
Find the point Q on the line $\displaystyle \frac{2 \mathrm{x}+4}{6}=\frac{\mathrm{y}+1}{2}=\frac{-2 \mathrm{z}+6}{-4}$ at a distance of $\displaystyle 3 \sqrt{2}$ from the point $\displaystyle \mathrm{P}(1,2,3)$.
Find the image of the point $\displaystyle (-1,5,2)$ in the line $\displaystyle \frac{2 \mathrm{x}-4}{2}=\frac{\mathrm{y}}{2}=\frac{2-\mathrm{z}}{3}$. Find the length of the line segment joining the points (given point and the image point).
Marking-scheme solution
The general point on the line $\displaystyle (3 \lambda-2,2 \lambda-1,2 \lambda+3)$ is $\displaystyle \mathbf{Q}$, from some $\displaystyle \lambda \in \mathbf{R}$
\[\mathrm{P} \mathrm{Q}=3 \sqrt{2} \Rightarrow(\mathrm{P} \mathrm{Q})^{2}=18 \Rightarrow(3 \lambda-3)^{2}+(2 \lambda-3)^{2}+(2 \lambda)^{2}=18
\]
Thus, the point is $\displaystyle \mathrm{Q}(-2,-1,3)$ or $\displaystyle \mathrm{Q}\left(\frac{56}{17}, \frac{43}{17}, \frac{111}{17}\right)$
Let $\displaystyle \mathbf{A}^{\prime}(\mathbf{a}, \mathbf{b}, \mathbf{c})$ be the image of the point $\displaystyle \mathbf{A}(-\mathbf{1}, \mathbf{5}, \mathbf{2})$ in the given line, also assume ' $\displaystyle M$ ' as the point of intersection of $\displaystyle A A^{\prime}$ with the given line, then ' $\displaystyle M$ ' is the mid-point of the line segment $\displaystyle \mathbf{A} \mathbf{A}^{\prime}$
And, $\displaystyle \mathbf{A A}^{\prime}=\sqrt{\mathbf{4 9 + 6 4 + 9}}=\sqrt{\mathbf{1 2 2}}$
Three Dimensional GeometryEquation of a Line in SpaceApplylong_answerhard
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CBSE Class 12 Mathematics past-paper question from the 2025board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.