We are given the equation:
$r \cos\theta \cdot \frac{1}{2} + r \sin\theta \cdot \frac{\sqrt{3}}{2} = 2$
Multiply through to simplify:
$\Rightarrow \frac{1}{2} r \cos\theta + \frac{\sqrt{3}}{2} r \sin\theta = 2$
Multiply both sides by 2:
$\Rightarrow r \cos\theta + \sqrt{3} r \sin\theta = 4$
Rewriting this equation:
$\Rightarrow \frac{r \cos\theta}{4} + \frac{r \sin\theta}{\frac{4}{\sqrt{3}}} = 1$
This is the equation of a straight line in polar form:
$\frac{r \cos\theta}{a} + \frac{r \sin\theta}{b} = 1$
Hence, the given equation represents a straight line.
Correct option: (D) a pair of straight lines
We are given:
$r \cos\left(\theta - \frac{\pi}{3}\right) = 2$
Using the identity: $\cos(A - B) = \cos A \cos B + \sin A \sin B$
$\Rightarrow r (\cos\theta \cdot \cos\frac{\pi}{3} + \sin\theta \cdot \sin\frac{\pi}{3}) = 2$
Substitute values: $\cos\frac{\pi}{3} = \frac{1}{2}, \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2}$
$\Rightarrow r \left(\cos\theta \cdot \frac{1}{2} + \sin\theta \cdot \frac{\sqrt{3}}{2}\right) = 2$
Multiply both sides by 2: $r \cos\theta + \sqrt{3} r \sin\theta = 4$
Use $x = r \cos\theta$, $y = r \sin\theta$
$\Rightarrow x + \sqrt{3} y = 4$
$\Rightarrow \frac{x}{4} + \frac{y}{\frac{4}{\sqrt{3}}} = 1$
This is the equation of a straight line.
Correct option: (D) a pair of straight lines
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