Question:
A plane passing through the point \((3, 1, 1)\) contains two lines whose direction ratios are \(1, -2, 2\) and \(2, 3, -1\) respectively. If this plane also passes through the point \((\alpha, -3, 5)\), then \(\alpha\) is equal to:
A plane passing through the point \((3, 1, 1)\) contains two lines whose direction ratios are \(1, -2, 2\) and \(2, 3, -1\) respectively. If this plane also passes through the point \((\alpha, -3, 5)\), then \(\alpha\) is equal to:
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The normal of the plane is the cross product of the two direction vectors.
Updated On: Nov 5, 2026
- \(-10\)
- \(10\)
- \(5\)
- \(-5\)
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The Correct Option is C
Solution and Explanation
Normal vector \(\vec n=(1,-2,2)\times(2,3,-1)\).
\(\vec n=\left((-2)(-1)-2\cdot3,\ 2\cdot2-1\cdot(-1),\ 1\cdot3-(-2)\cdot2\right)=(-4,5,7)\).
Plane through \((3,1,1)\): \(-4(x-3)+5(y-1)+7(z-1)=0\).
Substitute \((\alpha,-3,5)\): \(-4(\alpha-3)+5(-4)+7(4)=0\).
\(-4\alpha+12-20+28=0\Rightarrow -4\alpha+20=0\Rightarrow \alpha=5\).
Answer: (3) 5
\(\vec n=\left((-2)(-1)-2\cdot3,\ 2\cdot2-1\cdot(-1),\ 1\cdot3-(-2)\cdot2\right)=(-4,5,7)\).
Plane through \((3,1,1)\): \(-4(x-3)+5(y-1)+7(z-1)=0\).
Substitute \((\alpha,-3,5)\): \(-4(\alpha-3)+5(-4)+7(4)=0\).
\(-4\alpha+12-20+28=0\Rightarrow -4\alpha+20=0\Rightarrow \alpha=5\).
Answer: (3) 5
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