Question

I : Polar equation of $y=x\tan \alpha$ is $\theta =\alpha$.
II : Polar equation of $x^3=y^2(4-x)$ is $r\cos \theta =4\sin \theta$.
Which of the above statements is/are true?

• A. Only I
• B. Only II
• C. Both I and II
• D. Neither I nor II

Answer 1

Answer: (A)

Only I

We have $x=r\cos \theta ,y=r\sin \theta$
Statement 1: Given cartesian equation $y=x\tan \alpha$
$\Rightarrow r\sin \theta =r\cos \theta \tan \alpha$
$\Rightarrow \tan \theta =\tan \alpha$
$\Rightarrow \theta =\alpha$
Hence, statement I is true
Statement 2: $x^3=y^2(4-x)$
$\Rightarrow r^3{\cos }^3\theta -4r^2{\sin }^2\theta +r^3{\sin }^2\theta \cos \theta =0$

$\Rightarrow r^3\cos \theta ({\cos }^2\theta +{\sin }^2\theta )-4r^2{\sin }^2\theta =0$
$\Rightarrow r^3\cos \theta -4r^2{\sin }^2\theta =0$
$\Rightarrow r^2(r\cos \theta -4{\sin }^2\theta )=0$
$\Rightarrow r=0$ or $r\cos \theta =4{\sin }^2\theta$ or
Hence, statement II is not correct.
So, option A is correct.