Problem 46 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Find the number of integral ordered pairs \((x, y)\) satisfying the equation \(\tan ^{-1}\left(\frac{1}{x}\right)+\tan ^{-1}\left(\frac{1}{y}\right)=\tan ^{-1}\left(\frac{1}{10}\right)\).

Step-by-Step Solution

Answer

The number of integral ordered pairs \((x, y)\) satisfying the equation is 14.

Step 1: Simplify the Equation

Apply the formula (\(\tan^{-1}a+\tan^{-1}b = \tan^{-1}\left(\frac{a+b}{1-ab}\right)\)) to the equation. This results in: \(\tan^{-1} \left( \frac{ \frac{1}{x} + \frac{1}{y} }{ 1 - \frac{1}{10xy} } \right) = \tan^{-1}\left(\frac{1}{10}\right)\).

Step 2: Remove the inverse tangent function

Since the inverse tangent of both sides are equal, the values inside the brackets must be equal. Thus, we have: \(\frac{ \frac{1}{x} + \frac{1}{y} }{ 1 - \frac{1}{10xy} } = \frac{1}{10}\).

Step 3: Simplify the fractions

Multiply both sides with the denominator on the left to remove the fractions and simplify. This leads to:\(\frac{1}{x} + \frac{1}{y} = \frac{1}{10}(1 - \frac{1}{xy})\).

Step 4: Rearrange the equation

Multiply throughout by \(10xy\) to make the equation free of denominators. This results in: \(10y + 10x = xy - 1\).

Step 5: Rearrange to a quadratic equation form

Rearrange the equation into the standard form of a quadratic equation which ∃x is the variable: \(xy - 10x - 10y + 100 = 0\). This can be seen as a quadratic equation in x: \(yx^2 - 10x + 100 = 0\).

Step 6: Solving for x

This quadratic equation can be solved using the Quadratic formula \(x = \frac{-b±\sqrt{b^2-4ac}}{2a}\), where a = y, b = -10, and c = 100. Then we find:\(x = \frac{10±\sqrt{100-4*100*y}}{2y}\). Since x is an integer, within the square root, we must have a perfect square.

Step 7: Determine values for y

For the quantity inside the square root to be perfect square, \(100-4*100*y\) must be a perfect square. Let's denote \(100-4*100*y = k^2\), where k is an integer. Solve for y which results in \( y = \frac{100-k^2}{400}\). Since y is an integer, \(k^2\) must be of the form \(100+400*m\), where m is an integer.

Step 8: Possible values for k

Upon finding the possible values for k, through listing out those that satisfy the conditions, we'll find k in [-1, -2, -4, 0, 1, 2, 4]. For every value of k, there exist two values of x (because of the ± in the quadratic formula for x).

Step 9: Count the ordered pairs

Since there are 7 different numbers for k, and for each value of k, we have two solutions for x, so there are \(2 * 7 = 14\) solutions. Thus, there are 14 ordered pairs (x, y).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions help us find angles when we know the value of the trigonometric function. The problem uses \( \tan^{-1}\), which is the inverse of the tangent function. In general, when we have an equation like \( \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}(c)\), we can use a formula:

• \( a + b = c(1 - ab) \)

However, we need both angles, \(a\) and \(b\), to sit between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\).
This is key because it provides a straightforward way to add inverse tangents and equate expressions. By converting the original problem using this identity, it simplifies our task to handling a quadratic equation. This is a tactic often used in trigonometry to break down complex problems into more manageable parts.

Quadratic Equations

A quadratic equation is a second-degree polynomial, usually in the form \(ax^2 + bx + c = 0\). Solving quadratic equations is a cornerstone concept in math because they appear often in different contexts, including here with the x-values. Once we rewrote the problem using the \( \tan^{-1}\) formula, a quadratic form emerged: \( xy - 10x - 10y + 100 = 0 \).
To tackle this, we see the equation as related to x, specifically:

• \( x^2y - 10x + 100 = 0\)

We use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) to solve this. One critical requirement is that the expression under the square root, called the discriminant, must be a perfect square to ensure x yields integer solutions. This crucial step lets us find potential values for y and their corresponding x.

Integer Solutions

The solution must be integers because we're asked to find ordered pairs \( (x, y) \). After restructuring the quadratic equation, the task was to ensure that the discriminant becomes a perfect square. This is why we ended up focusing on \(100 - 4 \cdot 100 \cdot y\), setting it equal to another perfect square \(k^2\).

• We need \((100-k^2) = 400m\) for some integer \(m\).

Now, solving for possible values of y becomes dependent on finding integer values of \(k\) that satisfy the condition. Therefore, when y is calculated, there are integer values, which in return also gives integer solutions for x upon substitution back into the quadratic equation formula. This thoroughness ensures every potential pair \( (x, y) \) from solving \( \tan^{-1}\) problem are integers, leading to 14 total ordered pairs.