A root of the equation`17x^2 + 17x tan [2 tan^-1(1/5) - pi/4] - 10 = 0` is (i) `10/17` (ii)`-1` (iii)`-7/17` (iv)`1`

Options

A. `10/17`
B. `-1`
C. `-7/17`
D. 1

Explanation

To solve the equation \( 17x^2 + 17x \tan\left(2 \tan^{-1}\left(\frac{1}{5}\right) - \frac{\pi}{4}\right) - 10 = 0 \), we will follow these steps:

### Step 1: Simplify the expression inside the tangent function
We start with the expression \( \tan\left(2 \tan^{-1}\left(\frac{1}{5}\right) - \frac{\pi}{4}\right) \).

Using the identity \( \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \), we can express \( \tan(2 \tan^{-1}(x)) \):
- Let \( x = \frac{1}{5} \), then \( \tan\left(\tan^{-1}\left(\frac{1}{5}\right)\right) = \frac{1}{5} \).
- Therefore, \( \tan\left(2 \tan^{-1}\left(\frac{1}{5}\right)\right) = \frac{2 \cdot \frac{1}{5}}{1 - \left(\frac{1}{5}\right)^2} = \frac{\frac{2}{5}}{1 - \frac{1}{25}} = \frac{\frac{2}{5}}{\frac{24}{25}} = \frac{2 \cdot 25}{5 \cdot 24} = \frac{10}{24} = \frac{5}{12} \).
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