Question

If the sum of the roots of \(a x^{2}+b x+c=0\) is equal to the sum of the squares of their reciprocals then \(\mathrm{bc}^{2}, \mathrm{ca}^{2}, \mathrm{ab}^{2}\) are in (a) A.P (b) G,P (c) H.P (d) None of these

Short answer

The relationship between \(bc^2, ac^2\), and \(ab^2\) is an arithmetic progression (A.P).

Step-by-step solution

Find the sum of the roots

Using Vieta's formula, we can find the sum of the roots (\(S\)) of a quadratic equation \(ax^2 + bx + c = 0\). The sum is given by: \[S = \frac{-b}{a}\]

Find the sum of the squares of the reciprocals of the roots

Let the roots be \(r_1\) and \(r_2\). According to the given condition, the sum of their reciprocal squares is equal to the sum of the roots, i.e., \[\frac{1}{r_1^2} + \frac{1}{r_2^2} = S\]

Simplify the expression and find a relationship

Let's simplify the expression: \[\frac{1}{r_1^2} + \frac{1}{r_2^2} = \frac{r_1^2 + r_2^2}{r_1^2 r_2^2}\] Using Vieta's formula, we know that the product of the roots (\(P\)) for a quadratic equation \(ax^2 + bx + c = 0\): \[P = \frac{c}{a}\]. Thus, \[r_1 r_2 = \frac{c}{a}\] Now, substitute the values of S and P in the expression: \[\frac{r_1^2 + r_2^2}{(r_1 r_2)^2} = \frac{-b}{a}\] \[\frac{r_1^2 + r_2^2}{\left(\frac{c^2}{a^2}\right)} = \frac{-b}{a}\] Now multiply both sides by \(-a^3\): \[-a^3\left(\frac{r_1^2 + r_2^2}{\left(\frac{c^2}{a^2}\right)}\right) = -a^3 \times \frac{-b}{a}\] This gives us: \[bc^2 + ac^2 = ab^2\]

Determine the relationship

We got the relationship: \[bc^2 + ac^2 = ab^2\] Now let's check which of the given options meets this relationship. (a) A.P : In an arithmetic progression, the middle term would be the average of the first and last terms. So, if \(bc^2, ac^2\), and \(ab^2\) are in A.P, then \[2\times ac^2 = bc^2 + ab^2\], which is true. Therefore, the relationship is an A.P. So, the correct answer is: (a) A.P

Key concepts

Vieta's formulas

Vieta's formulas are a powerful set of equations that relate the coefficients of a polynomial to sums and products of its roots. For a quadratic equation of the form \(ax^2 + bx + c = 0\), Vieta's formulas state:

• The sum of the roots \((r_1 + r_2)\) is given by \(S = \frac{-b}{a}\).
• The product of the roots \((r_1 \times r_2)\) is \(P = \frac{c}{a}\).

These formulas are derived from the standard form of the quadratic equation and they simplify complex operations like solving for roots or performing algebraic manipulations involving roots.
In the given exercise, we use the sum of roots formula \(S = \frac{-b}{a}\) to help us solve the problem.

Arithmetic progression

An arithmetic progression (AP) is a sequence of numbers such that the difference between consecutive terms is constant. This difference is known as the 'common difference'. For example, in the sequence 2, 5, 8, 11, ..., each term increases by 3.
In the context of the exercise, we explore whether values \(bc^2, ac^2, ab^2\) form an arithmetic progression. For terms to be in AP, the difference between the second and first terms must equal the difference between the third and second terms:

• If our terms are in AP, we have \(2 \times ac^2 = bc^2 + ab^2\).

By verifying this condition, we can confirm the progression nature of the terms.

Reciprocal of roots

The reciprocal of the roots of a quadratic equation involves taking the inverse of each root. For roots \(r_1\) and \(r_2\), the reciprocals are \(\frac{1}{r_1}\) and \(\frac{1}{r_2}\). The exercise discusses the relationship between the sum of the roots and the sum of the squares of their reciprocals:

• The sum of the squares of reciprocals \(\frac{1}{r_1^2} + \frac{1}{r_2^2}\) must equal the sum of the roots \(S\).

Using simplifications and Vieta's formulas, the equation \(\frac{1}{r_1^2} + \frac{1}{r_2^2} = \frac{r_1^2 + r_2^2}{(r_1 r_2)^2}\) is formed.
This equation connects the concepts through mathematical manipulation.

Sum and product of roots

The sum and product of roots are fundamental concepts in solving quadratic equations. These are directly given by Vieta’s formulas:

• The sum of the roots is \(S = r_1 + r_2 = \frac{-b}{a}\).
• The product of the roots is \(P = r_1 \times r_2 = \frac{c}{a}\).

These formulas are crucial for quickly understanding relationships between a quadratic's coefficients and its roots.
In solving the exercise, the balance between the sum of the squares of the reciprocals and the sum of the roots involves both these formulas. They allow us to derive a relationship between \(bc^2, ac^2,\) and \(ab^2\) and check if they form an arithmetic progression, which is validated by the mathematical logic presented in the exercise.