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Mathematics · 2022 · 3 marks
CBSE 2022 · Region 5 · Set 2 · Q8
Find the equation of the plane passing through the line of intersection of the planes r ·( i + j + k )=10 and r ·(2 i +3 j - k )+4=0 and passing…
Find the equation of the plane passing through the line of intersection of the planes $\displaystyle \vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=10$ and $\displaystyle \vec{r} \cdot(2 \hat{i}+3 \hat{j}-\hat{k})+4=0$ and passing through the point $\displaystyle (-2,3,1)$.
Marking-scheme solution
Equation of plane through the intersection of given two planes is
\[\begin{aligned}
& \vec{r} \cdot[(\hat{i}+\hat{j}+\hat{k})+\lambda(2 \hat{i}+3 \hat{j}-\hat{k})]-10+4 \lambda=0 \\
& \vec{r} \cdot[(1+2 \lambda) \hat{i}+(1+3 \lambda) \hat{j}+(1-\lambda) \hat{k}]-10+4 \lambda=0
\end{aligned}
\]
Point (-$\displaystyle 2$, $\displaystyle 3$, $\displaystyle 1$) lies on it.
\[\begin{aligned}
& \therefore-2-4 \chi+3+9 \lambda+1-\lambda-10+4 \chi=0 \\
& 8 \lambda=8 \Rightarrow \lambda=1
\end{aligned}
\]
∴ Equation of plane is \(\displaystyle \vec{r} .(3 \hat{i}+4 \hat{j})-6=0\)
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CBSE Class 12 Mathematics past-paper question from the 2022 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.