Question

The cartesian co-ordinates of the point whose polar co-ordinates are as $(13,\pi -{\tan }^{-1}\left(\frac{5}{12}\right))$, are

• A. $(12,5)$
• B. $(-12,5)$
• C. $(-12,-5)$
• D. $(12,-5)$

Answer 1

The correct option is B $(-12,5)$

Given , $(13,\pi -{\tan }^{-1}\left(\frac{5}{12}\right))$
$r=13,\theta =\pi -{\tan }^{-1}\left(\frac{5}{12}\right)$
$x^2+y^2=169$ ...(1)
Also, ${\tan }^{-1}\left(\frac{y}{x}\right)=\pi -{\tan }^{-1}\left(\frac{5}{12}\right)$
$\Rightarrow \frac{y}{x}=\tan (\pi -{\tan }^{-1}\left(\frac{5}{12}\right))$
$\Rightarrow \frac{y}{x}=\tan \left({\tan }^{-1}\left(\frac{5}{12}\right)\right)$

$\Rightarrow y=\frac{5}{12}x$
So, by (1), $x^2=144$
$\Rightarrow x=12,-12$
$\Rightarrow y=5,-5$
But since, $\theta$ lies in second quadrant
So, $x=-12,y=5$
Hence, the cartesian form is $(-12,5)$.