Let `x_1,x_2,x_3,x_4` be four non zero numbers satisfying the equation `tan^-1 (a/x)+tan^-1(b/x)+tan^-1(c/x)+tan^-1(d/x)=pi/2` then which ofthe following relation(s) hold good?

Options

A. `sum_(i=1)^(4)x_(i)=a+b+c+d`
B. `sum_(i=1)^(4)(1)/(x_(i))=0`
C. `prod_(i=1)^(4)x_(i)=abcd`
D. `(x_(1)+x_(2)+x_(3))(x_(2)+x_(3)+x_(4))(x_(3)+x_(4)+x_(1))(x_(4)+x_(1)+x_(2))=abcd`

Answer

To solve the equation

\[ \tan^{-1}\left(\frac{a}{x}\right) + \tan^{-1}\left(\frac{b}{x}\right) + \tan^{-1}\left(\frac{c}{x}\right) + \tan^{-1}\left(\frac{d}{x}\right) = \frac{\pi}{2}, \]

we will use the property of the tangent inverse function and some algebraic manipulations.

### Step 1: Use the property of tangent inverse
We know that

\[ \tan^{-1}(A) + \tan^{-1}(B) = \tan^{-1}\left(\frac{A + B}{1 - AB}\right), \]

provided \(AB < 1\). We will apply this property iteratively.

### Step 2: Combine the first two terms
Let \(A = \frac{a}{x}\) and \(B = \frac{b}{x}\). Then,

\[ \tan^{-1}\left(\frac{a}{x}\right) + \tan^{-1}\left(\frac{b}{x}\right) = \tan^{-1}\left(\frac{\frac{a}{x} + \frac{b}{x}}{1 - \frac{ab}{x^2}}\right) = \tan^{-1}\left(\frac{a + b}{x} \cdot \frac{x^2}{x^2 - ab}\right). \]

### Step 3: Combine the next two terms
Now, let \(C = \frac{c}{x}\) and \(D = \frac{d}{x}\). Similarly, we have:

\[ \tan^{-1}\left(\frac{c}{x}\right) + \tan^{-1}\left(\frac{d}{x}\right) = \tan^{-1}\left(\frac{c + d}{x} \cdot \frac{x^2}{x^2 - cd}\right). \]

### Step 4: Combine the results from Steps 2 and 3
Now we combine the results of Steps 2 and 3:

\[ \tan^{-1}\left(\frac{a + b}{x} \cdot \frac{x^2}{x^2 - ab}\right) + \tan^{-1}\left(\frac{c + d}{x} \cdot \frac{x^2}{x^2 - cd}\right) = \frac{\pi}{2}. \]

Using the property again, we can write:

\[ \tan^{-1}\left(\frac{\frac{(a + b)x + (c + d)x}{x^2}}{1 - \frac{(a + b)(c + d)}{x^2}}\right) = \frac{\pi}{2}. \]

### Step 5: Set the denominator to zero
For the tangent inverse to equal \(\frac{\pi}{2}\), the denominator must equal zero:

\[ 1 - \frac{(a + b)(c + d)}{x^2} = 0. \]

This leads to:

\[ x^2 = (a + b)(c + d). \]

### Step 6: Analyze the polynomial
Now, we can analyze the polynomial formed by the roots \(x_1, x_2, x_3, x_4\). The polynomial can be expressed as:

\[ x^4 - (ab + cd + ac + ad + bc + bd)x^2 + abcd = 0. \]

### Step 7: Find the sum and product of roots
Using Vieta's formulas, we find:

- The sum of the roots \(x_1 + x_2 + x_3 + x_4 = 0\) (since the coefficient of \(x^3\) is zero).
- The product of the roots \(x_1 x_2 x_3 x_4 = abcd\).

### Conclusion
Thus, the relations that hold good are:

1. \(x_1 + x_2 + x_3 + x_4 = 0\)
2. \(x_1 x_2 x_3 x_4 = abcd\)