Problem 236 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \frac{a x}{\cos \theta}+\frac{b y}{\sin \theta}=a^{2}-b^{2} \text { and } \frac{a x \sin \theta}{\cos ^{2} \theta}-\frac{b y \cos \theta}{\sin ^{2} \theta}=0, \text { show that }(a x)^{\frac{2}{3}}+(b y)^{\frac{2}{3}}=\left(a^{2}-b^{2}\right)^{\frac{2}{3}} \text { . } $$

Step-by-Step Solution

Answer

The given mathematical equations imply that \( (ax)^{2/3} + (by)^{2/3} = (a^{2}-b^{2})^{2/3} \). This is concluded by systematically simplifying and re-arranging the equations and then substituting the derived expressions for x and y into the required proof to show equality.

Step 1: Simplify the given equations

Firstly, simplify the given equations. Divide the second equation by \( \sin \theta / \cos ^{2} \theta \), which yields:\( ax - bytan \theta = 0 \)From this point, it can be inferred that:\( ax = bytan \theta \)

Step 2: Substitute into first equation

Next, substitute \( ax = bytan \theta \) into the first equation:\( \frac{a(bytan \theta)}{\cos \theta}+\frac{b y}{\sin \theta}=a^{2}-b^{2} \)This simplifies to:\( aby+by = a^{2}-b^{2} \)After further simplification, it yields:\( by(a+\sin \theta) = a^{2}-b^{2} \)Solving for \( y \), we get:\( y = \frac{a^{2}-b^{2}}{b(a+\sin \theta)} \)

Step 3: Substitute y into ax = bytan θ

Substitute \( y = \frac{a^{2}-b^{2}}{b(a+\sin \theta)} \) into \( ax = bytan \theta \). This gives:\( ax = \frac{a^{2}-b^{2}}{a+\sin \theta} \cdot tan \theta \)Solving for \( x \), we get:\( x = \frac{a(a^{2}-b^{2})}{(a+\sin \theta)a} \)

Step 4: Substitute x and y into the required proof

Substitute \( x = \frac{a(a^{2}-b^{2})}{(a+\sin \theta)a} \) and \( y = \frac{a^{2}-b^{2}}{b(a+\sin \theta)} \) into the expression \( (ax)^{2/3} + (by)^{2/3} \). If it simplifies to \( (a^{2}-b^{2})^{2/3} \), then it proves the given equation correct.

Key Concepts

Equation Simplification

When tackling any differential calculus problems, simplifying the equations should be your first step. This means breaking down complex expressions into simpler forms to make them more manageable and understandable.

Here is how you can do it effectively:

• Identify common terms in the equation. For instance, both the given equations in the problem have terms involving sin and cos.
• Look for opportunities to factor out common elements from individual terms or the whole equation. This can reduce complexity significantly.
• If possible, condense terms into single terms. This often involves using basic arithmetic operations like addition and subtraction.

In the provided solution, the first and second equations were simplified to reach a manageable form such as \( ax - by\tan \theta = 0 \). Breaking equations down in such a way makes subsequent steps, such as substitution or solving for unknowns, far easier.

Trigonometric Identities

Trigonometric identities are useful tools that help in solving equations involving trigonometric functions. They express relationships between sine, cosine, tangent, and other trigonometric functions.

Several key identities might assist you in this exercise:

• The Pythagorean identities, such as \( \sin^2 \theta + \cos^2 \theta = 1 \), which often help simplify complex expressions.
• Tangent can be expressed in terms of sine and cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This is crucial when your equations involve \( \tan \theta \), as seen in our problem.
• Other identities include secant, cosecant, and their various transformations. Including these can often provide alternative pathways to solving a problem.

By understanding and applying these identities, you can significantly reduce the effort needed to rework trigonometric parts of any mathematical proof.

Algebraic Manipulation

Algebraic manipulation involves reshaping equations or expressions to extract useful relationships or solve for a particular variable. It often involves both arithmetical operations and strategic substitutions.

Here’s how you can leverage algebraic manipulation effectively:

• Use multiplication or division across the equation to eliminate fractions or decimals, like dividing the entire equation by \( \sin \theta / \cos^2 \theta \) in this problem.
• Substitute known values or identities wherever possible. We substituted \( ax = by\tan \theta \) into another equation to simplify our solving process.
• Work systematically to isolate variables of interest. The main goal is to express everything in terms of one variable and simplify from there.

Through these manipulative techniques, you arrive at simpler forms like \( x = \frac{a(a^2-b^2)}{(a+\sin \theta)a} \) which can then be substituted into the final expression to demonstrate the required condition.