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3D Geometry question

2024 · 27 Jan · Shift 1 · Q43
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  5. /2024 · 27 Jan · Shift 1 · Q43

3D Geometry question

2024 · 27 Jan · Shift 1 · Q43

JEE MainMathematics3D GeometryMCQ+4 / −1
If the shortest distance between the lines x−41=y+12=z−3\frac{x-4}{1}=\frac{y+1}{2}=\frac{z}{-3}1x−4​=2y+1​=−3z​ and x−λ2=y+14=z−2−5\frac{x-\lambda}{2}=\frac{y+1}{4}=\frac{z-2}{-5}2x−λ​=4y+1​=−5z−2​ is 65\frac{6}{\sqrt{5}}5​6​, then the sum of all possible values of λ\lambdaλ is :
  1. A
    10
  2. B
    5
  3. C
    7
  4. D
    8
View written solutionFree

Correct answer: D

  1. Write the lines in vector form

For x−41=y+12=z−3,\frac{x-4}{1}=\frac{y+1}{2}=\frac{z}{-3},1x−4​=2y+1​=−3z​, we take a point and direction vector as A=(4,−1,0),d⃗1=(1,2,−3).A=(4,-1,0), \qquad \vec d_1=(1,2,-3).A=(4,−1,0),d1​=(1,2,−3).

For x−λ2=y+14=z−2−5,\frac{x-\lambda}{2}=\frac{y+1}{4}=\frac{z-2}{-5},2x−λ​=4y+1​=−5z−2​, we take B=(λ,−1,2),d⃗2=(2,4,−5).B=(\lambda,-1,2), \qquad \vec d_2=(2,4,-5).B=(λ,−1,2),d2​=(2,4,−5).

  1. Use the formula for shortest distance between two skew lines

The shortest distance is D=∣(AB→)⋅(d⃗1×d⃗2)∣∣d⃗1×d⃗2∣.D=\frac{|(\overrightarrow{AB})\cdot(\vec d_1\times \vec d_2)|}{|\vec d_1\times \vec d_2|}.D=∣d1​×d2​∣∣(AB)⋅(d1​×d2​)∣​.

Here, AB→=B−A=(λ−4,0,2).\overrightarrow{AB}=B-A=(\lambda-4,0,2).AB=B−A=(λ−4,0,2).

  1. Compute the cross product
\begin{vmatrix} \hat i & \hat j & \hat k \\ 1 & 2 & -3 \\ 2 & 4 & -5 \end{vmatrix}$$ $$=\hat i(2\cdot(-5)-(-3)\cdot4)-\hat j(1\cdot(-5)-(-3)\cdot2)+\hat k(1\cdot4-2\cdot2).$$ $$=\hat i(-10+12)-\hat j(-5+6)+\hat k(4-4)$$ $$=(2,-1,0).$$ So, $$|\vec d_1\times \vec d_2|=\sqrt{2^2+(-1)^2}=\sqrt{5}.$$ --- 4. **Compute the scalar triple product** $$\overrightarrow{AB}\cdot(\vec d_1\times \vec d_2)=(\lambda-4,0,2)\cdot(2,-1,0)=2(\lambda-4).$$ Hence, $$D=\frac{|2(\lambda-4)|}{\sqrt{5}}.$$ Given that $$D=\frac{6}{\sqrt{5}},$$ so $$\frac{|2(\lambda-4)|}{\sqrt{5}}=\frac{6}{\sqrt{5}}.$$ Therefore, $$|2(\lambda-4)|=6$$ $$|\lambda-4|=3.$$ So, $$\lambda=7 \quad \text{or} \quad \lambda=1.$$ --- 5. **Find the sum of all possible values** $$1+7=8.$$ --- 6. **Check options** The correct option is: $$\boxed{\text{D: }8}$$ --- 7. **Compare with stored correct answer** Stored correct answer is **D**, which matches our result.
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