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Question

If the line $$\displaystyle \frac{x - 1}{1} = \frac{y + 1}{-2} = \frac{z + 1}{\lambda}$$ lies in the plane $$\displaystyle 3x - 2y + 5z = 0$$ then $$\displaystyle \lambda$$ is


A
1
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B
75
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C
57
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D
no possible value
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Solution

The correct option is D $$\displaystyle -\frac{7}{5}$$
We have equation of plane,
$$3x-2y+5z=0.......(1)$$

We have line,

$$\dfrac{x-1}{1}=\dfrac{y+1}{-2}=\dfrac{z+1}{\lambda}=\mu......(2)$$
General point on line is,
$$P=(\mu+1,-2\mu-1,\lambda\mu-1)$$
Since line (2) lies on line plane (1),so point P satisfy equation (1)
Therefore,
$$3(\mu+1)-2(2\mu-1)+5(\lambda\mu-1)=0$$
$$3\mu+3+4\mu+2+5\mu\lambda-5=0$$
$$7\mu+5\mu\lambda=0$$
$$\lambda=\dfrac{-7}{5}$$ 
Therefore option (B) is Correct.


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