Question

If $(\alpha ,\beta )$ is a point of intersection of the lines $x\cos \theta +y\sin \theta =3\text{and}x\sin \theta -y\cos \theta =4$ where $\theta$ is parameter, then maximum value of $2^{\frac{\alpha +\beta }{\sqrt{2}}}$ is

• A. $16$
• B. $32$
• C. $64$
• D. $128$
• E. $256$

Answer 1

Answer: (D)

$128$

$\left(1\right)\Rightarrow (x\cos \theta +y\sin \theta =3)\times \cos \theta$

$\left(2\right)\Rightarrow (x\sin \theta -y\cos \theta =4)\times \sin \theta$

$\Rightarrow x{\cos }^2\theta +y\sin \theta \cos \theta =3\cos \theta$

$xsi{n}^2\theta -y\sin \theta \cos \theta =4\sin \theta$

$\Rightarrow x({\sin }^2\theta +{\cos }^2\theta )=3\cos \theta +4\sin \theta$

$\Rightarrow x=3\cos \theta +4\sin \theta =\alpha$

Put $\alpha$ in (1)

$\alpha \cos \theta +y\sin \theta =3$

$\Rightarrow 3{\cos }^2\theta +4\sin \theta \cos \theta +y\sin \theta =3$

$\Rightarrow y\sin \theta =3{\sin }^2\theta +4\sin \theta \cos \theta$

$\Rightarrow y=3\sin \theta +4\cos \theta =\beta$

$\alpha +\beta =3(\sin \theta +\cos \theta )+4(\cos \theta +\sin \theta )$

$=7(\sin \theta +\cos \theta )$

$2^{(\alpha +\beta )/\sqrt{2}}=2^{7(\sin \theta +\cos \theta )/\sqrt{2}}$

$Max\left(2^{(\alpha +\beta )/\sqrt{2}}\right)=Max\left(2^{7(\frac{1}{\sqrt{2}}\sin \theta +\frac{1}{\sqrt{2}}\cos \theta )}\right)$

$=max\left(2^{7\sin (\theta +\pi /4)}\right)$

$=2^{7\times 1}=2^7=128$