JEE Main 2022 · 25 Jun · Shift 2 · Mathematics Q30

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MathematicsInteger+4 / -1Question 30 of 90

Let $l_1$ be the line in $xy$ -plane with $x$ and $y$ intercepts $\dfrac18$ and $\dfrac1{4\sqrt2}$ respectively, and $l_2$ be the line in $zx$ -plane with $x$ and $z$ intercepts $-\dfrac18$ and $-\dfrac1{6\sqrt3}$ respectively. If $d$ is the shortest distance between the line $l_1$ and $l_2$ , then $d^{-2}$ is equal to ______.

Your answer

Answer and worked solution

Official answer: 51

Lines. $l_1$ through $\left(\frac18,0,0\right)$ with direction $(1,-\sqrt2,0)$ ; $l_2$ through $\left(-\frac18,0,0\right)$ with direction $\left(1,0,-\frac4{3\sqrt3}\right)$ .

Common normal. $\vec n\propto\left(1,\frac1{\sqrt2},\frac{3\sqrt3}4\right)$ , with $|\vec n|^2=1+\frac12+\frac{27}{16}=\frac{51}{16}$ .

Distance. $d=\dfrac{\left|\left(-\frac14,0,0\right)\cdot\vec n\right|}{|\vec n|}=\dfrac{1/4}{\sqrt{51}/4}=\dfrac1{\sqrt{51}}$ , so $d^{-2}=51$ .

Answer: 51

Topics:Skew lines Equation of a line in space

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JEE Main 2022 · 25 Jun · Shift 2, Mathematics question 30