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9.Straight Line
hard
રેખાઓ $x \cos \theta+y \sin \theta=7, \theta \in\left(0, \frac{\pi}{2}\right)$ ના યામાક્ષો વચ્યેની રેખાખંડોના મધ્યબિંદુઓ દ્વારા આલેખાયેલ વક્ર પર બિંદુ $\left(\alpha, \frac{7 \sqrt{3}}{3}\right)$ આવેલ હોય, તો $\alpha=.........$
A
$7$
B
$-7$
C
$-7 \sqrt{3}$
D
$7 \sqrt{3}$
(JEE MAIN-2023)
Solution
$\operatorname{pt}\left(\alpha, \frac{7 \sqrt{3}}{3}\right)$
$x -\text { intercept }=\frac{7}{\cos \theta}$
$y -\text { intercept }=\frac{7}{\sin \theta}$
$A:\left(\frac{7}{\cos \theta}, 0\right) B :\left(0, \frac{7}{\sin \theta}\right)$
Locus of mid pt $M 🙁 h , k )$
$h =\frac{7}{2 \cos \theta}, k =\frac{7}{2 \sin \theta}$
$\frac{7}{2 \sin \theta}=\frac{7 \sqrt{3}}{3} \Rightarrow \sin \theta=\frac{\sqrt{3}}{2} \Rightarrow \theta=\frac{\pi}{3}$
$\alpha=\frac{7}{2 \cos \theta}=7$
Standard 11
Mathematics
