The equation \(r^2=cos^2(\theta-\frac{\pi}{3})=2\) represents
- a parabola
- a hyperbola
- a circle
- a pair of straight line
The Correct Option is D
Approach Solution - 1
\(r\,cos\theta.\frac{1}{2}+r\,sin\theta.\frac{\sqrt3}{2}=2\)
\(r\,cos\theta+\sqrt3 r\,sin\theta=4\)
\(\Rightarrow \frac{r\,cos\theta}{4}+\frac{r\,sin\theta}{\frac{4}{\sqrt3}}=1\)
Hence, This is a straight line
So, the correct option is (D) : a pair of straight line.
Approach Solution -2
\(r\cos(\theta-\frac{\pi}{3})=2\)
\(⇒r(\cos\theta.\cos\frac{\pi}{3}+\sin\theta.\sin\frac{\pi}{3})=2\)
\(⇒r(\cos\theta.\frac{1}{2}+\sin\theta.\frac{\sqrt3}{2})=2\)
\(⇒(r\cos\theta)+\sqrt3.(r\sin\theta)=4\)
\(⇒x+\sqrt3y=4\)
\(⇒(\frac{x}{4}+\frac{y}{\sqrt3}=1)\) → Represents a straight line
So, the correct option is (D) : a pair of straight line.
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Concepts Used:
Derivatives of Functions in Parametric Forms
The derivative of a function in parametric form is emanated in two parts; the first derivative and the second derivative. To emanate the equation, let us suppose there are two dependent variables x and y, and one independent variable ‘t’.
Therefore, x = (x)t, and y = (y)t
Example for First Derivative:
Example for Second Derivative:
