Q85 / 200 · TS EAMCET 2001 · Mathematics

Equation of curve in polar coordinates is $\frac{l}{r}=2 \sin ^2 \frac{\theta}{2}$, then it represents

  • A
    a straight line
  • Correct answer
    B
    a parabola
  • C
    a circle
  • D
    an ellipse

Solution

We have, $\frac{l}{r}=2 \sin ^2 \frac{\theta}{2} \Rightarrow \frac{l}{r}=1-\cos \theta$ $\begin{array}{rlrl}\Rightarrow & l & =r(1-\cos \theta) \Rightarrow l+r \cos \theta=r \\ \Rightarrow & & r^2 & =(l+r \cos \theta)^2 \\ \Rightarrow & x^2+y^2 & =(l+x)^2 \\ \Rightarrow & x^2+y^2 & =l^2+x^2+2 l x \\ \Rightarrow & & y^2 & =2 l\left(x+\frac{l}{2}\right)\end{array}$ Which represents a parabola.

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

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