Q65 / 160 · TS EAMCET 2006 · Mathematics

If $5 \cos x+12 \cos y=13$, then the maximum value of $5 \sin x+12 \sin y$ is :

  • A
    12
  • Correct answer
    B
    $\sqrt{120}$
  • C
    $\sqrt{20}$
  • D
    13

Solution

$\because 5 \cos x+12 \cos y=13$ $\Rightarrow \quad(5 \cos x+12 \cos y)^2=169$ $\therefore(5 \cos x+12 \cos y)^2+(5 \sin x+12 \sin y)^2$ $\Rightarrow \quad(13)^2+(5 \sin x+12 \sin y)^2$ $=25+144+120(\sin x \sin y+\cos x \cos y)$ $\Rightarrow \quad(5 \sin x+12 \sin y)^2=120 \cos (x-y)$ $\because \quad-1 \leq \cos (x-y) \leq 1$ $\therefore$ Maximum value of $5 \sin x+12 \sin y=\sqrt{120}$

Hence, the correct answer is option: 'B'.

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