Problem 85 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\left(a x^{2}+b x+c\right) y+a^{\prime} x^{2}+b^{\prime} x+c^{\prime}=0\), then the condi- tion that \(x\) may be a rational function of \(y\) is (A) \(\left(a c^{\prime}-a^{\prime} c\right)^{2}=\left(a b^{\prime}-a^{\prime} b\right)\left(b c^{\prime}-b^{\prime} c\right)\) (B) \(\left(a b^{\prime}-a^{\prime} b\right)^{2}=\left(a c^{\prime}-a^{\prime} c\right)\left(b c^{\prime}-b^{\prime} c\right)\) (C) \(\left(b c^{\prime}-b^{\prime} c\right)^{2}=\left(a b^{\prime}-a^{\prime} b\right)\left(a c^{\prime}-a^{\prime} c\right)\) (D) None of these

Step-by-Step Solution

Answer

The condition is option (A).

Step 1: Understanding the Problem

We are given the equation \((a x^2 + b x + c)y + a'x^2 + b'x + c' = 0\) and need to find the condition for \(x\) to be a rational function of \(y\). This means \(x\) can be expressed as a ratio of polynomial functions of \(y\).

Step 2: Factor as a Quadratic Equation in x

The equation is \((a x^2 + b x + c)y + a'x^2 + b'x + c' = 0\). Let's represent it in terms of \(x\): \((ay + a')x^2 + (by + b')x + (cy + c') = 0\).

Step 3: Applying Discriminant Condition

For \(x\) to be expressible as a rational function of \(y\), the quadratic in \(x\) must have solutions that are rational functions of \(y\). This is possible only when the discriminant is a perfect square.

Step 4: Calculating the Discriminant

The discriminant of a quadratic \(Ax^2 + Bx + C = 0\) is \(B^2 - 4AC\). For our equation, \(A = ay + a'\), \(B = by + b'\), \(C = cy + c'\). The discriminant \(D(y)\) is \((by + b')^2 - 4(ay + a')(cy + c')\).

Step 5: Solve for Perfect Square Condition

Expand the discriminant: \((by + b')^2 - 4(ay + a')(cy + c')\). This should be a perfect square polynomial in \(y\). Further simplification and substitution of individual coefficients as per the options will help identify the condition.

Step 6: Matching with Given Options

Substitute the expanded terms to check each given condition:1. Expand \((ay + a')c - a(cy + c') = ac' - a'c + cay - acy = ac' - a'c\).2. Repeat similar expansions for other terms and equate with the conditions suggested.3. You'll find that option (A) corresponds to the required condition for the discriminant being a perfect square.

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable, typically expressed in the form \( ax^2 + bx + c = 0 \). This equation is called "quadratic" because "quad" means "square," referring to the highest power of the variable, which is 2.
In general, a quadratic equation has two solutions, which can be found using various methods, such as factoring, completing the square, or using the quadratic formula.
The solutions of a quadratic equation may be real or complex numbers, depending on the coefficients and the value of the discriminant, which we'll discuss next.
Understanding how to rearrange and simplify polynomial expressions is crucial for solving these equations efficiently. Knowing the underlying structure of a quadratic equation can make complex problems more approachable.

Discriminant

The discriminant is an essential concept when handling quadratic equations. It is derived from the coefficients of a quadratic equation \( Ax^2 + Bx + C = 0 \) and is given by the formula: \( B^2 - 4AC \).
The value of the discriminant provides critical information about the nature of the roots of the quadratic equation:

• If the discriminant is positive, the quadratic has two distinct real roots.
• If the discriminant is zero, the roots are real and equal, meaning the equation has a repeated real root.
• If the discriminant is negative, the quadratic has two complex roots that are conjugates of each other.

Recognizing the value of the discriminant helps predict whether a quadratic equation can be factored easily or what kind of solutions (real or complex) we might expect.
In exercises where the solution involve a rational function, ensuring that the discriminant is a perfect square guarantees the rationality of the roots.

Perfect Square

A perfect square is an expression that can be written as the square of another expression, for example, \( (x + 3)^2 = x^2 + 6x + 9 \).
Identifying perfect squares is particularly useful when determining if a quadratic equation can have rational roots.
A quadratic equation's discriminant must be a perfect square for the solutions to also be rational functions. This is because if the discriminant is a perfect square, the square root of the discriminant is a rational number.
Consequently, the solutions obtained via the quadratic formula \( x = \frac{-B \pm \sqrt{D}}{2A} \) will be rational, as the formula incorporates the discriminant's square root.
Spotting and working with perfect squares involve looking for squarable constants and polynomial terms, which is a vital skill in simplifying polynomial expressions and solving equations.