JEE Main 2025 Jan 22 Shift 2, Mathematics Q23: Heights and Distance
JEE Main2025Jan 22, Shift 2Mathematics
Q.
Let the distance between two parallel lines be 5 units and a point lie between the lines at a unit distance from one of them. An equilateral triangle is formed such that lies on one of the parallel lines, while lies on the other. Then is equal to ________.
Correct answer: 28
Solution
$ PR=\text{cosec}\theta,\qquad PQ=4\sec(30^\circ+\theta) $
$ \text{For an equilateral triangle} $
$ d=PR=PQ $
$ \Rightarrow \cos(\theta+30^\circ)=4\sin\theta $
$ \Rightarrow \dfrac{\sqrt{3}}{2}\cos\theta - \dfrac{1}{2}\sin\theta = 4\sin\theta $
$ \Rightarrow \tan\theta = \dfrac{1}{3\sqrt{3}} $
$ QR^2=d^2=\text{cosec}^2\theta=28 $
Concept behind this question
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