Question

The no. of solution in \(x \in \left(-\frac{1}{2\sqrt{6}}, \frac{1}{2\sqrt{6}}\right)\) of equation \(\tan^{-1}4x + \tan^{-1}6x = \frac{\pi}{6}\) is :

• A. 0
• B. 1
• C. 2
• D. 3

Hint:

When using the formula for \(\tan^{-1}A + \tan^{-1}B\), always check the condition \(AB<1\). The given interval often provides a hint about which form of the formula to use.

Answer 1

The Correct Option is B

Step 1: Problem Understanding:
We are tasked with solving the trigonometric equation involving inverse tangents: \[ \tan^{-1}(4x) + \tan^{-1}(6x) = \frac{\pi}{6} \] and finding the number of solutions for \(x\) within a specific interval. 

Step 2: Key Formula or Approach: 
Using the identity for the sum of two inverse tangent functions: \[ \tan^{-1}(A) + \tan^{-1}(B) = \tan^{-1}\left( \frac{A+B}{1 - AB} \right) \quad \text{for} \quad AB<1. \] Here, \( A = 4x \) and \( B = 6x \), so the product \( AB = 24x^2 \). We require \( 24x^2<1 \), which simplifies to \( x^2<\frac{1}{24} \). Therefore, \( |x|<\frac{1}{\sqrt{24}} = \frac{1}{2\sqrt{6}} \). 

Step 3: Detailed Explanation: 
Applying the formula to the given equation: \[ \tan^{-1}(4x) + \tan^{-1}(6x) = \tan^{-1}\left( \frac{4x+6x}{1 - (4x)(6x)} \right) = \tan^{-1}\left( \frac{10x}{1 - 24x^2} \right) \] We are given that this equals \( \frac{\pi}{6} \): \[ \tan^{-1}\left( \frac{10x}{1 - 24x^2} \right) = \frac{\pi}{6}. \] Taking the tangent of both sides: \[ \frac{10x}{1 - 24x^2} = \tan\left( \frac{\pi}{6} \right) = \frac{1}{\sqrt{3}}. \] Now, we solve for \(x\): \[ 10\sqrt{3}x = 1 - 24x^2, \] \[ 24x^2 + 10\sqrt{3}x - 1 = 0. \] This is a quadratic equation in \(x\). Using the quadratic formula: \[ x = \frac{-10\sqrt{3} \pm \sqrt{(10\sqrt{3})^2 - 4(24)(-1)}}{2(24)}. \] Simplifying: \[ x = \frac{-10\sqrt{3} \pm \sqrt{300 + 96}}{48} = \frac{-10\sqrt{3} \pm \sqrt{396}}{48} = \frac{-10\sqrt{3} \pm 6\sqrt{11}}{48}. \] Thus, the two possible solutions are: \[ x_1 = \frac{-5\sqrt{3} + 3\sqrt{11}}{24}, \quad x_2 = \frac{-5\sqrt{3} - 3\sqrt{11}}{24}. \] 

Step 4: Checking the Solutions against the Interval: 
The interval is \( \left( -\frac{1}{2\sqrt{6}}, \frac{1}{2\sqrt{6}} \right) \). \[ \frac{1}{2\sqrt{6}} \approx 0.204. \] Checking the solutions: \[ x_1 \approx 0.0538 \quad (\text{which lies within the interval}). \] \[ x_2 \approx -0.775 \quad (\text{which lies outside the interval}). \] Therefore, there is only one solution within the interval, which is \(x_1\).