JEE Main 2025 — Trigonometric Equations Question with Solution
JEE Main 2025 (7 Apr Shift 1)
Question
If for , the points lie on , then is equal to :
Choose an option
Show full solutionCorrect option: D
Correct answer
D75
Step-by-step explanation
$\begin{aligned}
& x=3\left(\frac{\tan \theta+\sqrt{3}}{1-\sqrt{3} \tan \theta}\right) \\ & x-\sqrt{3} \tan \theta=3 \tan \theta+3 \sqrt{3} \\ & \tan \theta=\frac{x-3 \sqrt{3}}{3+\sqrt{3} x} \ldots(1) \\ & 2\left(\frac{\tan \theta+\frac{1}{\sqrt{3}}}{1-\frac{\tan \theta}{\sqrt{3}}}=y\right) \\ & 2(\sqrt{3} \tan \theta+1)=y(\sqrt{3}-\tan \theta) \ldots .
\end{aligned}\begin{aligned}
& 2\left(\frac{x-3 \sqrt{3}}{\sqrt{3}+x}+1\right)=y\left(\sqrt{3}-\frac{(x-3 \sqrt{3})}{\sqrt{3}(\sqrt{3}+x)}\right) \\ & 2 \sqrt{3}(x-3 \sqrt{3}+x+\sqrt{3})=y(3(\sqrt{3}+x)-x+3 \sqrt{3}) \\ & 4 \sqrt{3} x-12=y(2 x+6 \sqrt{3}) \\ & x y-2 \sqrt{3} x+3 \sqrt{3} y-6=0 \\ & \Rightarrow \alpha=-2 \sqrt{3}, \beta=3 \sqrt{3}, \gamma=-6 \\ & \alpha^2+\beta^2+\gamma^2=12+27+36=75
\end{aligned}$
& x=3\left(\frac{\tan \theta+\sqrt{3}}{1-\sqrt{3} \tan \theta}\right) \\ & x-\sqrt{3} \tan \theta=3 \tan \theta+3 \sqrt{3} \\ & \tan \theta=\frac{x-3 \sqrt{3}}{3+\sqrt{3} x} \ldots(1) \\ & 2\left(\frac{\tan \theta+\frac{1}{\sqrt{3}}}{1-\frac{\tan \theta}{\sqrt{3}}}=y\right) \\ & 2(\sqrt{3} \tan \theta+1)=y(\sqrt{3}-\tan \theta) \ldots .
\end{aligned}\begin{aligned}
& 2\left(\frac{x-3 \sqrt{3}}{\sqrt{3}+x}+1\right)=y\left(\sqrt{3}-\frac{(x-3 \sqrt{3})}{\sqrt{3}(\sqrt{3}+x)}\right) \\ & 2 \sqrt{3}(x-3 \sqrt{3}+x+\sqrt{3})=y(3(\sqrt{3}+x)-x+3 \sqrt{3}) \\ & 4 \sqrt{3} x-12=y(2 x+6 \sqrt{3}) \\ & x y-2 \sqrt{3} x+3 \sqrt{3} y-6=0 \\ & \Rightarrow \alpha=-2 \sqrt{3}, \beta=3 \sqrt{3}, \gamma=-6 \\ & \alpha^2+\beta^2+\gamma^2=12+27+36=75
\end{aligned}$
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This is a previous-year question from JEE Main 2025, covering the Trigonometric Equations chapter of Mathematics. PrepSharp catalogues every PYQ from JEE Main with a verified answer key and step-by-step solution prepared by IIT alumni — so you can search by chapter, topic or year and revise efficiently.