JEE Advanced 2019 · Paper 1 · Mathematics Q12

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MathematicsMultiple correct+4 / -1Question 48 of 54

Let $L_1$ and $L_2$ denote the lines

$\vec r = \hat i + \lambda(-\hat i + 2\hat j + 2\hat k),\ \lambda\in\mathbb R\quad\text{and}\quad\vec r = \mu(2\hat i - \hat j + 2\hat k),\ \mu\in\mathbb R$

respectively. If $L_3$ is a line which is perpendicular to both $L_1$ and $L_2$ and cuts both of them, then which of the following options describe(s) $L_3$ ?

\vec r = \frac29(4\hat i + \hat j + \hat k) + t(2\hat i + 2\hat j - \hat k),\ t\in\mathbb R

\vec r = \frac29(2\hat i - \hat j + 2\hat k) + t(2\hat i + 2\hat j - \hat k),\ t\in\mathbb R

\vec r = \frac13(2\hat i + \hat k) + t(2\hat i + 2\hat j - \hat k),\ t\in\mathbb R

\vec r = t(2\hat i + 2\hat j - \hat k),\ t\in\mathbb R

Answer and worked solution

Official answer: A, B, C

Direction of $L_3$ .
$(-1,2,2)\times(2,-1,2)=(6,6,-3)\parallel(2,2,-1).$

Feet of the common perpendicular. Let $P_1=(1-\lambda,2\lambda,2\lambda)$ on $L_1$ and $P_2=(2\mu,-\mu,2\mu)$ on $L_2$ . Requiring $P_1-P_2$ to be perpendicular to both lines gives $9\lambda=1$ and $9\mu=2$ . So
$P_1=\tfrac29(4,1,1),\qquad P_2=\tfrac29(2,-1,2).$

Checking each point.

(A) passes through $P_1$ and (B) through $P_2$ , both with direction $(2,2,-1)$ . ✓
(C) passes through $\frac13(2,0,1)$ , the midpoint of $P_1P_2$ , which also lies on $L_3$ . ✓
(D) would need the origin on $L_3$ . From $P_1+t(2,2,-1)$ , $x=0$ gives $t=-\frac49$ , and then $y=\frac29-\frac89\ne0$ . ✗

(A), (B), (C)

Topics:Skew lines Cross product

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