Question:

The equation \(r^2=cos^2(\theta-\frac{\pi}{3})=2\) represents

Updated On: May 10, 2025
  • a parabola
  • a hyperbola
  • a circle
  • a pair of straight line
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The Correct Option is D

Approach Solution - 1

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

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Approach Solution -2

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

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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: