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← JEE Main 2022 · 27 Jul · Shift 2
MathematicsSingle correct+4 / -1Question 15 of 90

JEE Main 2022 · 27 Jul · Shift 2, Mathematics question 15

If the line of intersection of the planes ax+by=3ax+by=3 and ax+by+cz=0ax+by+cz=0, a>0a>0 makes an angle 30° with the plane y−z+2=0y-z+2=0, then the direction cosines of the line are :

Choose one option
Answer and worked solution

Official answer: A

Direction. (a,b,0)×(a,b,c)∥(b,−a,0)(a,b,0)\times(a,b,c)\parallel(b,-a,0).

Angle with the plane. sin⁡30∘=a2a2+b2\sin30^\circ=\frac{a}{\sqrt2\sqrt{a^2+b^2}} gives b=±ab=\pm a. So the direction cosines are ±(12,−12,0)\pm\left(\frac1{\sqrt2},-\frac1{\sqrt2},0\right) or ±(12,12,0)\pm\left(\frac1{\sqrt2},\frac1{\sqrt2},0\right).

Both (A) and (B) fit, and NTA's key accepts either. (B)

Topics:Angles and distancesDirection cosines and ratios
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