If line x-12=y+13=z-14 and x-31=y-λ2=z1 intersect each other, then λ=.….

MHT CET · Maths · Three Dimensional Geometry

If line $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4}$ and $\frac{x - 3}{1} = \frac{y - λ}{2} = \frac{z}{1}$ intersect each other, then $λ = . \dots .$

A $\frac{7}{2}$
B $\frac{3}{2}$
C $\frac{9}{2}$
D $\frac{5}{2}$

Verified Solution

Answer & Solution

Correct Answer

(C) $\frac{9}{2}$

Step-by-step Solution

Detailed explanation

Given lines are

\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4} = λ_{1} (l e t)

….. (i)

a n d \frac{x - 3}{1} = \frac{y - λ}{2} = \frac{z}{1} = λ_{2} (l e t)

…… (ii) is
Then, any point on line (i) is

(2 λ_{1} + 1,3 λ_{1} - 1,4 λ_{1} + 1)

and any point on line (ii) is

(λ_{2} + 3,2 λ_{2} + λ, λ_{2})

Clearly, then lines (i) and (ii) will intersect if

(2 λ_{1} + 1,3 λ_{1} - 1,4 λ_{1} + 1)

(λ_{2} + 3,2 λ_{2} + λ, λ_{2})

For some particular value of

λ_{1}

and

λ_{2}

\Rightarrow 2 λ_{1} + 1 = λ_{2} + 3,3 λ_{1} - 1 = 2 λ_{2} + λ

a n d 4 λ_{1} + 1 = λ_{2}

\Rightarrow 2 λ_{1} - λ_{2}, 3 λ_{1} - 2 λ_{2} = λ + 1

and

4 λ_{1} - λ_{2} = - 1

We get

λ_{1} = - \frac{3}{2}

and

λ_{1} = - 5

Now, putting the values of

λ_{1}

and

λ_{2}

3 λ_{1} - 2 λ_{2} = λ + 1

\Rightarrow 3 (\frac{- 3}{2}) - 2 (- 5) = λ + 1 \Rightarrow \frac{- 9}{2} + 10 λ + 1

\Rightarrow λ + 1 = \frac{11}{2} \Rightarrow λ = \frac{9}{2}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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