QJEEAdv19Maf081: JEE Advanced 2019 Mathematics - Class 12 JEEAdv19Maf081 Solution: Let L 1 and L 2 denote the lines , R and respectively. If L 3…

Question

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Let L₁ and L₂ denote the lines

r = \widehat i + \lambda ( - \widehat i + 2\widehat j + 2\widehat k)

,

\lambda

\in

R

and

r = \mu (2\widehat i - \widehat j + 2\widehat k),\,\mu \in R

respectively. If L₃ is a line which is perpendicular to both L₁ and L₂ and cuts both of them, then which of the following options describe(s) L₃?

A $r = {2 \over 9}(2\widehat i - \widehat j + 2\widehat k) + t(2\widehat i + 2\widehat j - \widehat k),\,t \in R$
B $r = {1 \over 3}(2\widehat i + k) + t(2\widehat i + 2\widehat j - \widehat k),\,t \in R$
C $r = {2 \over 9}(4\widehat i + \widehat j + \widehat k) + t(2\widehat i + 2\widehat j - \widehat k),\,t \in R$
D r = $t(2\widehat i + 2\widehat j - \widehat k)$ , $t \in R$

Accepted Answer

Given lines

{L_1}:r = \widehat i + \lambda ( - \widehat i + 2\widehat j + 2\widehat k),\,\lambda \in R

and

{L_2}:r = \mu (2\widehat i - \widehat j + 2\widehat k),\,\mu \in R

and since line L₃ is perpendicular to both lines L₁ and L₂.

Then a vector along L₃ will be,

| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ - 1 & 2 & 2 \\ 2 & - 1 & 2 \end{matrix} | = \hat{i} (4 + 2) - \hat{j} (- 2 - 4) + \hat{k} (1 - 4)

= 6\widehat i + 6\widehat j - 3\widehat k = 3(2\widehat i + 2\widehat j - \widehat k)

....(i)

Now, let a general point on line L₁.

P(1 - \lambda ,\,2\lambda ,\,2\lambda )

and on line L₂ as

Q(2\mu - \mu ,\,2\mu )

and let P and Q are point of intersection of lines L₁, L₃ and L₂, L₃, so direction ratio's of L₃

(2\mu + \lambda - 1,\, - \mu - 2\lambda ,\,\,2\mu - 2\lambda )

....(ii)

Now,

{{2\mu + \lambda - 1} \over 2} = {{\, - \mu - 2\lambda } \over 2}\, = {{\,2\mu - 2\lambda } \over 1}

[from Eqs. (i) and (ii)]

\Rightarrow \lambda = {1 \over 9}

and

\mu = {2 \over 3}

So,

P\left( {{8 \over 9},{2 \over 9},{2 \over 9}} \right)

and

Q\left( {{4 \over 9}, - {2 \over 9},{4 \over 9}} \right)

Now, we can take equation of line L₃ as

r = a + t(2\widehat i + 2\widehat j - \widehat k)

, where a is position vector of any point on line L₃ and possible vector of a are

\left( {{8 \over 9}\widehat i + {2 \over 9}\widehat j + {2 \over 9}\widehat k} \right)

or

\left( {{4 \over 9}\widehat i - {2 \over 9}\widehat j + {4 \over 9}\widehat k} \right)

or

\left( {{2 \over 3}\widehat i + {1 \over 3}\widehat k} \right)

Hence, options (a), (b) and (c) are correct.

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