If the sum of the roots of the quadratic equation ax² + bx + c = 0 is equal to the sum of the square

Q: Wait, why do we divide by $abc$ in the final step? Is there a rule for this?

That is a fantastic observation! In JEE problems involving ratios of coefficients, dividing by the product of the variables is a classic technique. It transforms a homogeneous equation into a form that reveals the progression structure. It is not just a random step; it is a strategic move to isolate the ratios $a/b, b/c,$ and $c/a$ that define the progression.

Q: The problem asks for $a/c, b/a, c/b$ in H.P., but my derivation gave me an A.P. Did I make a mistake?

You are actually on the right track! You likely derived that $b/c, a/b, c/a$ are in A.P. (or a similar sequence). Remember the definition of Harmonic Progression: if a sequence $x, y, z$ is in H.P., then their reciprocals $1/x, 1/y, 1/z$ form an A.P. You have found the A.P. sequence; now you just need to invert those terms to confirm the H.P. relationship.

Q: Can I just assume $a=1, b=2, c=3$ and test values to find the answer?

While testing values is a clever exam-day strategy for multiple-choice questions, it is risky. The algebraic derivation is the 'JEE way' because it proves the property for *all* valid quadratic equations, not just one specific case. Always aim for the derivation to ensure your logic is bulletproof.

Question

If the sum of the roots of the quadratic equation $ax^2 + bx + c = 0$ is equal to the sum of the squares of their reciprocals, then $a/c, b/a$ and $c/b$ are in

Options:

(A) Arithmetic - Geometric Progression

(B) Arithmetic Progression

(C) Geometric Progression

(D) Harmonic Progression.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before a classic quadratic equation: $ax^2 + bx + c = 0$. It looks simple, almost unassuming. But within its roots, $\alpha$ and $\beta$, lies a hidden geometry of numbers.

Today, we are going to peel back the layers of this equation to uncover a beautiful relationship between its coefficients.

### The Bridge: Vieta's Formulas

Our journey begins with the fundamental connection between the roots and the coefficients. We know that the sum of the roots is:

$$\alpha + \beta = -\frac{b}{a}$$

And the product is:

$$\alpha\beta = \frac{c}{a}$$

These are our building blocks. The problem presents us with a fascinating condition: the sum of the roots is equal to the sum of the squares of their reciprocals. Mathematically, this is:

$$\alpha + \beta = \frac{1}{\alpha^2} + \frac{1}{\beta^2}$$

### The Algebraic Transformation

To make sense of the right-hand side, we need to simplify it. By taking the common denominator, we get:

$$\frac{\alpha^2 + \beta^2}{(\alpha\beta)^2}$$

Now, we face a challenge: we don't know $\alpha^2 + \beta^2$ directly. But wait! We have an identity for that: $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$.

Substituting this, our equation becomes:

$$\alpha + \beta = \frac{(\alpha + \beta)^2 - 2\alpha\beta}{(\alpha\beta)^2}$$

This is the moment where the abstract becomes concrete. We substitute our Vieta's relations:

$$-\frac{b}{a} = \frac{(-\frac{b}{a})^2 - 2(\frac{c}{a})}{(\frac{c}{a})^2}$$

### The Algebraic Dance

Now, let's simplify. The right side becomes:

$$\frac{\frac{b^2}{a^2} - \frac{2c}{a}}{\frac{c^2}{a^2}}$$

By finding a common denominator in the numerator, we get $\frac{b^2 - 2ac}{a^2}$. When we divide this by the denominator $\frac{c^2}{a^2}$, the $a^2$ terms cancel out beautifully, leaving us with:

$$-\frac{b}{a} = \frac{b^2 - 2ac}{c^2}$$

Cross-multiplying gives us $-bc^2 = ab^2 - 2a^2c$. Rearranging this, we find:

$$2a^2c = ab^2 + bc^2$$

### The Final Reveal

We are almost there. We have a relation, but we need to see the progression. The master trick is to divide the entire equation by $abc$:

$$\frac{2a^2c}{abc} = \frac{ab^2}{abc} + \frac{bc^2}{abc}$$

This simplifies to:

$$\frac{2a}{b} = \frac{b}{c} + \frac{c}{a}$$

This is the classic condition for an Arithmetic Progression: $2y = x + z$. Thus, $\frac{b}{c}, \frac{a}{b}, \frac{c}{a}$ are in A.P.

Since the terms $\frac{a}{c}, \frac{b}{a}, \frac{c}{b}$ are the reciprocals of the terms in an A.P., they must form a **Harmonic Progression**. And there it is: the hidden harmony of the quadratic equation.

Animated Solution for Mathematics - Quadratic Equations: If the sum of the roots of the quadratic equation $a x^{2} + b x + c = 0$ is equal to the sum of the squares of their reciprocals, then $a / c, b / a$ and $c / b$ are in

Select Answer:

Visualized Solution

Define Roots and Basic Relations

Express the Given Condition

Simplify the Right Hand Side

Apply Algebraic Identity

Substitute Coefficients

Simplify Numerator and Denominator

Cancel Denominators

Cross Multiplication

Rearrange Terms

The Crucial Division

Reveal Arithmetic Progression

Final Conclusion: Harmonic Progression

The Sigma Insight: Relation Between Roots and Coefficients

Analyzing the Setup

The Bridge

The Algebraic Transformation

The Algebraic Dance

The Final Reveal

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