Question

In an polar curve r=f (θ) what is the relation between θ & the coordinates (x,y)?

(a) tan⁡θ = \(\frac{x}{y} \)

(b) (1+sin⁡θ) = \(\frac{y}{x} \)

(c) (1+sec^2 θ) = \(\frac{y^2}{x^2} \)

(d) (1+cos⁡θ) = \(\frac{x}{y} \)

I have been asked this question in an international level competition.

The question is from Polar Curves topic in division Differential Calculus of Engineering Mathematics

Answer 1

Right option is (c) (1+sec^2 θ) = \(\frac{y^2}{x^2} \)

To elaborate: w.k.t for the polar curve r=f (θ) x=rcos⁡θ, y=rsin⁡θ

dividing them we get \(\frac{y}{x} = \frac{r sin⁡θ}{r cos⁡θ} = tan⁡θ\)

squaring on both side \(\frac{y^2}{x^2} = tan^2 θ = (1+sec^2 θ)\).