Question

The eccentricity of the conic \( \frac{5}{r} = 2 + 3\cos\theta + 4\sin\theta \) is

• A. $\frac{4}{2}$
• B. 1
• C. $\frac{5}{2}$
• D. 0

Hint:

$a\cos \theta + b\sin \theta$ can be written as $\sqrt{a^2+b^2}\cos(\theta - \phi)$.

Answer 1

The Correct Option is C

Step 1: Simplify the Conic Form
Combine the trig terms: $3\cos \theta +4\sin \theta = 5(\frac{3}{5}\cos \theta +\frac{4}{5}\sin \theta) = 5\cos(\theta - \phi)$ where $\cos \phi = 3/5$.
Step 2: Compare with Standard Form
$\frac{5}{r} = 2 + 5\cos(\theta - \phi) \Rightarrow \frac{5/2}{r} = 1 + \frac{5}{2}\cos(\theta - \phi)$. Standard polar form is $\frac{l}{r} = 1 + e\cos(\theta - \phi)$.
Step 3: Identify Eccentricity
Comparing, we get $e = 5/2$.
Final Answer: (c)