Question

The number of solutions of equation $5\sec \theta -13=12\tan \theta$ in $[0,2\pi ]$ is

• A. 2
• B. 1
• C. 4
• D. 0

Answer 1

The correct option is A 2

$5sec\theta -13=12tan\theta$

$\Rightarrow 12sin\theta +13cos\theta =5$

$\Rightarrow \frac{12}{\sqrt{313}}sin\theta +\frac{13}{\sqrt{313}}cos\theta =\frac{5}{\sqrt{313}}$

Let $cos\alpha =\frac{13}{\sqrt{313}}\Rightarrow sin\alpha =\frac{12}{\sqrt{313}}$

$\Rightarrow cos\theta .cos\alpha +sin\theta .sin\alpha =\frac{5}{\sqrt{313}}$

$\Rightarrow cos(\theta -\alpha )=\frac{5}{\sqrt{313}}$

Hence solution in the given interval is,

$\theta =\alpha -co{s}^{-1}\left(\frac{5}{\sqrt{313}}\right),\alpha +co{s}^{-1}\left(\frac{5}{\sqrt{313}}\right)$

Hence, option 'A' is correct.