Problem 110 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ y=\sqrt{(a-x)(x-b)}-(a-b) \tan ^{-1} \sqrt{\frac{a-x}{x-b}} $$

Step-by-Step Solution

Answer

In summary, the function \(y=\sqrt{(a-x)(x-b)}-(a-b)\tan^{-1}\sqrt{\frac{a-x}{x-b}}\) is a combination of a square root term, a subtraction term, and the tangent inverse of a fraction. The square root term represents changes in x relative to constants a and b, while the tangent inverse, or arctangent, represents the angle formed between the x-axis and a right-angled triangle formed by the square root. The function provides insights into relationships between angles, positive and negative changes in x, and the parameters a and b.

Step 1: Understand the components of the function

The function is composed of the following parts: 1. The square root: \(\sqrt{(a-x)(x-b)}\) 2. Tangent inverse: \(\tan^{-1}\) 3. The fraction inside the tangent inverse: \( \sqrt{\frac{a-x}{x-b}}\) Now let's break down and understand each component.

Step 1: Square Root #(a-x)(x-b)

Consider the square root part: \(\sqrt{(a-x)(x-b)}\). There are two factors inside the square root: (a-x) and (x-b). These factors represent changes in x relative to the constants a and b.

Step 2: Understand the Tangent Inverse

The tangent inverse function \(\tan^{-1}(x)\) is also known as the arctangent function. It is used to find the angle whose tangent is x. In our given function, the tangential part represents the angle formed between the x-axis and right-angled triangle formed by the square root.

Step 3: Fraction inside the Tangent Inverse

Consider the fraction inside the tangent inverse: \(\sqrt{\frac{a-x}{x-b}}\). This fraction represents the ratio of (a-x) to (x-b). It can be interpreted as a transformation of the given function. The square root of this fraction is what is being passed into \(\tan^{-1}\).

Step 4: Putting it all together

Now that we have broken down the components of the function, let's put them all together again to understand it holistically: \(y=\sqrt{(a-x)(x-b)}-(a-b)\tan^{-1}\sqrt{\frac{a-x}{x-b}}\) The function is a combination of a square root term and a subtraction term involving the tangent inverse of a fraction. The function represents a combination of geometric transformations, and can provide insights into relationships between angles, positive and negative changes in x, and the parameters a and b.

Key Concepts

Square Root Function

The square root function is vital in understanding mathematical relationships. In the expression \( \sqrt{(a-x)(x-b)} \), it helps us determine how values of \( x \) relate to constants \( a \) and \( b \). Here, the function consists of two parts, \( (a-x)\) and \( (x-b) \). These indicate how far \( x \) is from both endpoints. By placing these inside the square root, we measure a non-linear scaling or distortion.

When using square roots, remember:

• They return the principal (non-negative) square root.
• The expression inside must be non-negative to yield real numbers.
• In graphs, they create a curved output, representing natural growth or decay.

Understanding square roots mathematically enhances your grasp on how functions curve and change with different inputs. The visual of these changes depicts the scaling effect in geometric shapes.

Inverse Trigonometric Functions

Inverse trigonometric functions help find angles from given trigonometric ratios. The \( \tan^{-1} \) or arctangent is especially useful for determining angles given a ratio, often between -\( \frac{\pi}{2} \) and \( \frac{\pi}{2} \). In the component \( (a-b) \tan^{-1} \sqrt{\frac{a-x}{x-b}} \), \( \tan^{-1} \) takes the square root term, representing an angle in a triangle formed by our geometric change.

Consider these points:

• Arctangent returns an angle where tangent has the specified value.
• Being bounded between -\( \frac{\pi}{2} \) and \( \frac{\pi}{2} \), it helps maintain angle validity in equations.
• It’s used in inverse processes, solving from result back to original angle.

In equations like ours, the tangent inverse offers insights into the angles formed by expressions, rendering the geometry more comprehensible.

Function Transformation

Function transformation involves shifting, stretching, or reflecting functions to produce a new curve. In the equation, transformations involve both algebraic manipulations and geometric translations. The square root and arctangent are combined in transformative ways to alter the identity of \( y \) versus \( x \).

Key transformations here include:

• Shift: Adjusting slopes and angles with parameters \( a \) and \( b \), influencing horizontal translations.
• Scale: Changes due to multiplicative factors like \( a-b \) scale outputs, adjusting vertical stretch.
• Rotation/Reflection: Through arctangent and square root relations changing shapes formed by graphs.

Understanding these transformations aids in visualizing the evolution and shift in function plots, crucial for advanced studies in calculus.

Geometric Interpretation

Geometric interpretation is the visualization of algebraic expressions to grasp their meaning fully. For the function \( y = \sqrt{(a-x)(x-b)} - (a-b) \tan^{-1} \sqrt{\frac{a-x}{x-b}} \), consider the half of a hyperbola shape for the square root part and an angular interpretation for \( \tan^{-1} \).

Important geometric insights:

• Triangle Angles: The \( \tan^{-1} \) section relates to angles in triangles formed by changes in \( x \).
• Symmetry & Asymptotes: The components create symmetry, and potential asymptotes highlight important values where functions increase or become undefined.
• Curve Behavior: Non-linear outputs dictate how curves bend and shape over specific intervals.

Visualizing these elements helps connect the algebraic solutions with their geometric counterparts, creating a full picture of how functions behave in complex systems.