JEE Main 2022 · 30 Jul · Shift 2 · Mathematics Q87

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MathematicsInteger+4 / -1Question 87 of 90

Let L $_1$ be a line in the yz - plane with y and z intercepts $\dfrac14$ and $\dfrac1C$ (C > 0) respectively. Let L $_2$ be the line in the xz - plane with x and z intercepts $\dfrac13$ and $-\dfrac1C$ respectively. If the shortest distance between L $_1$ and L $_2$ is $\dfrac15$ , then C $^2$ is equal to _______.

Your answer

Answer and worked solution

Official answer: 75

Lines. $L_1$ through $\left(0,\frac14,0\right)$ with direction $\left(0,-\frac14,\frac1C\right)$ ; $L_2$ through $\left(\frac13,0,0\right)$ with direction $\left(-\frac13,0,-\frac1C\right)$ .

Scaled directions $(0,-C,4)$ and $(-C,0,-3)$ , with cross product $(3C,-4C,-C^2)$ of length $C\sqrt{C^2+25}$ . Joining vector $\left(\frac13,-\frac14,0\right)$ gives dot product $2C$ .

Distance $\dfrac2{\sqrt{C^2+25}}=\dfrac15$ , so $C^2=75$ .

Answer: 75

Topics:Skew lines

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JEE Main 2022 · 30 Jul · Shift 2, Mathematics question 87