Problem 58 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the quadratic equation whose roots are reciprocal of the roots of the equation \(a x^{2}+b x+c=0\).

Step-by-Step Solution

Answer

The quadratic equation whose roots are the reciprocals of the roots of the equation \(ax^2 + bx + c = 0\) is \(Ax^2 - bx + ac = 0\).

Step 1: Write expressions for the sum of roots of the given equation and the new equation

For the given equation, the sum of roots \(p + q\) is given by \(-\frac{b}{a}\). For the new equation, the sum of reciprocal roots \(\frac{1}{p} + \frac{1}{q}\), can be simplified into \(\frac{p+q}{pq}\). Step 2: Find the product of the roots

Step 2: Write expressions for the product of roots of the given equation and the new equation

For the given equation, the product of roots \(pq\) is given by \(\frac{c}{a}\). For the new equation, the product of reciprocal roots is \(\frac{1}{p} \cdot \frac{1}{q} = \frac{1}{pq}\). Step 3: Relate the expressions to their conjugates

Step 3: Express the sum and product of reciprocal roots in terms of the sum and product of the original roots

We have the sum of reciprocal roots as \(\frac{p+q}{pq}\) and the product of reciprocal roots as \(\frac{1}{pq}\). Now, from Step 1, we know that the sum of roots p and q is \(-\frac{b}{a}\), and from Step 2, the product of roots p and q is \(\frac{c}{a}\). Therefore, the sum and product of the reciprocal roots can be written as \(\frac{-b}{ac}\) and \(\frac{a}{c}\) respectively. Step 4: Write the new quadratic equation

Step 4: Use the expressions for the sum and product of reciprocal roots to construct the new quadratic equation

Knowing that the sum and product of the roots can be related to the coefficients of the equation, let the new quadratic equation we are trying to find be \(Ax^2 + Bx + C = 0\). The sum of reciprocal roots \(\frac{-B}{A} = \frac{-b}{ac}\) and the product of reciprocal roots \(\frac{C}{A} = \frac{a}{c}\). Step 5: Solve for coefficients A, B, and C

Step 5: Determine A, B, and C from the relationships above

From the relationship \(\frac{-B}{A} = \frac{-b}{ac}\), we get B = -b. And from the relationship \(\frac{C}{A} = \frac{a}{c}\), we get C = ac. Therefore, the new quadratic equation is \(Ax^2 - bx + ac = 0\). In summary, the quadratic equation whose roots are the reciprocals of the roots of the equation \(ax^2 + bx + c = 0\) is \(Ax^2 - bx + ac = 0\).

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in a single variable x with a coefficient that isn't zero, represented in the standard form as \(ax^2 + bx + c = 0\), where 'a', 'b', and 'c' are constants. To solve such equations for x, various methods can be used, including factoring, completing the square, using the quadratic formula, or graphing.

The graphical representation of a quadratic equation is a parabola, which may have 0, 1, or 2 real roots depending on the discriminant \(b^2 - 4ac\). These roots are the x-intercepts of the graph and represent the values of x that satisfy the equation.

Sum and Product of Roots

In the realm of algebra, there are useful relationships between the coefficients of a quadratic equation and its roots. For an equation \(ax^2 + bx + c = 0\), the sum of the roots, usually denoted as \(p + q\), is \(-\frac{b}{a}\) and the product of the roots, represented as \(pq\), is \(\frac{c}{a}\).

These relationships stem from the factoring of the quadratic equation and are handy for quickly calculating the roots' sum and product without needing to solve the entire equation. They are ultimately a consequence of the way polynomials split into linear factors.

Reciprocal Roots

Reciprocal roots are a pair of numbers such that the product of the pair is 1, meaning if 'p' and 'q' are roots of a quadratic equation \(ax^2 + bx + c = 0\), then their reciprocals are \(\frac{1}{p}\) and \(\frac{1}{q}\). The reciprocal roots are important for conceptualizing inverses and for solving equations that involve proportional relationships.

Understanding reciprocal roots can also facilitate the process of finding a new quadratic equation whose roots are the reciprocals of another. This interrelationship provides a deep insight into the symmetrical nature of roots in polynomial equations.

Algebraic Identities

Algebraic identities are mathematical equations that express an equality and hold true for all values of the variables within a certain range of applicability. In the context of quadratic equations, one of the fundamental identities is the expansion \((x + p)(x + q) = x^2 + (p + q)x + pq\), which relates the coefficients of the quadratic to the sum and product of its roots.

These identities are significant as they provide shortcuts for complicated algebraic computations and help in solving polynomial equations more efficiently. They play a pivotal role in constructing new equations, such as finding a quadratic equation given the roots or in this case, given the reciprocal of the roots.