Question

Which of the following statements are true?
(A) The curve r = a(1 + cos\(\theta\)) is symmetrical about the initial line
(B) The curve r = 2(1 - 2 sin\(\theta\)) is symmetrical about the initial line
(C) The curve r = a(1 + sin\(\theta\)) is symmetrical about the line \(\theta\) = \(\pi\)/2
(D) The curve r = a sin(3\(\theta\)) is symmetrical about the initial line
(E) The curve r² = a²cos(2\(\theta\)) is symmetrical about pole
Choose the correct answer from the options given below:

• A. (A), (B) and (D) only
• B. (A), (C), (D) and (E) only
• C. (A), (B), (D) and (E) only
• D. (A), (C) and (E) only

Hint:

Quick rules for polar symmetry:
If the equation only contains \(\cos\theta\), it's symmetric about the initial line (x-axis).
If the equation only contains \(\sin\theta\), it's symmetric about \(\theta=\pi/2\) (y-axis).
If \(r\) appears as \(r^2\) or \(\theta\) appears as a multiple within a function (like \(\cos(n\theta)\)), it often has pole symmetry.
These are useful shortcuts, but always apply the formal tests (\(\theta \to -\theta\), etc.) if unsure.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question tests the rules for symmetry of curves given in polar coordinates \(r = f(\theta)\).
Symmetry about the initial line (\(\theta=0\), the x-axis): Occurs if replacing \(\theta\) with \(-\theta\) leaves the equation unchanged.
Symmetry about the line \(\theta=\pi/2\) (the y-axis): Occurs if replacing \(\theta\) with \(\pi-\theta\) leaves the equation unchanged.
Symmetry about the pole (origin): Occurs if replacing \(r\) with \(-r\) OR replacing \(\theta\) with \(\theta+\pi\) leaves the equation unchanged.
Step 2: Detailed Explanation:

(A) r = a(1 + cos\(\theta\)): Replace \(\theta\) with \(-\theta\). Since \(\cos(-\theta) = \cos(\theta)\), the equation becomes \(r = a(1 + \cos\theta)\), which is unchanged. So, it is symmetrical about the initial line. Statement (A) is true.
(B) r = 2(1 - 2 sin\(\theta\)): Replace \(\theta\) with \(-\theta\). Since \(\sin(-\theta) = -\sin(\theta)\), the equation becomes \(r = 2(1 - 2(-\sin\theta)) = 2(1+2\sin\theta)\), which is different. So, it is not symmetrical about the initial line. Statement (B) is false.
(C) r = a(1 + sin\(\theta\)): Replace \(\theta\) with \(\pi-\theta\). Since \(\sin(\pi-\theta) = \sin(\theta)\), the equation becomes \(r = a(1 + \sin\theta)\), which is unchanged. So, it is symmetrical about the line \(\theta=\pi/2\). Statement (C) is true.
(D) r = a sin(3\(\theta\)): Replace \(\theta\) with \(-\theta\). Since \(\sin(-3\theta) = -\sin(3\theta)\), the equation becomes \(r = -a\sin(3\theta)\), which is different. So, it is not symmetrical about the initial line. Statement (D) is false. (It is symmetric about \(\theta = \pi/2\)).
(E) r² = a²cos(2\(\theta\)): To test for symmetry about the pole, replace \(r\) with \(-r\). The equation becomes \((-r)^2 = a^2\cos(2\theta)\), which is \(r^2 = a^2\cos(2\theta)\), unchanged. So, it is symmetrical about the pole. Statement (E) is true.
Step 3: Final Answer:
The true statements are (A), (C), and (E). This corresponds to option (D).