For p, q ∈ R, consider the real valued function f(x) = (x-p)² - q, x ∈ R and q 0. Let a₁, a₂, a₃ a

Q: Wait, why can't $f(a_i)$ be $500$ for all $i$? If $|f(a_i)| = 500$, shouldn't they all be the same?

That is a classic trap! The absolute value condition $|f(a_i)| = 500$ allows $f(a_i)$ to be either $500$ or $-500$. Because the parabola is not a flat line, the function values change as you move away from the vertex. The points closer to the vertex ($a_2, a_3$) will have different values than those further away ($a_1, a_4$).

Q: Why did we define the A.P. terms as $p \pm d$ and $p \pm 3d$? Is that allowed?

Absolutely! In fact, it is the most elegant way to handle symmetry. Since the mean is $p$, the terms must be symmetric about $p$. Defining them this way eliminates the variable $p$ immediately, saving you from messy calculations.

Q: How do we know for sure that $9d^2 - q$ is the positive $500$ and $d^2 - q$ is the negative $500$?

Great observation. Since $d^2 > 0$, it is mathematically certain that $9d^2 > d^2$. Therefore, $9d^2 - q$ is strictly greater than $d^2 - q$. If both were $500$, then $9d^2 = d^2$, which implies $d=0$ (not an A.P.). If both were $-500$, then $9d^2 = d^2$ again. Thus, the only logical possibility is the larger expression being positive and the smaller being negative.

Question

For $p, q \in \mathbb{R}$, consider the real valued function $f(x) = (x-p)^2 - q, x \in \mathbb{R}$ and $q > 0$. Let $a_1, a_2, a_3$ and $a_4$ be in an arithmetic progression with mean $p$ and positive common difference. If $|f(a_i)| = 500$ for all $i = 1, 2, 3, 4$, then the absolute difference between the roots of $f(x) = 0$ is ________.

Sigma Percentile · Accepted Answer

### Analyzing the Geometry of the Parabola

Imagine you are standing on a coordinate plane, looking at the graph of $f(x) = (x-p)^2 - q$. Because the coefficient of the squared term is positive, this is an upward-opening parabola. It has a vertex at $(p, -q)$ and is perfectly symmetric about the vertical line $x = p$.

In the world of JEE Advanced, whenever you see a quadratic expression in the form $(x-p)^2 - q$, do not rush to expand it. Instead, embrace the vertex form, as it reveals the function's behavior relative to its axis of symmetry.

Our goal is to find the absolute difference between the roots of $f(x) = 0$. Setting $f(x) = 0$ yields $(x-p)^2 = q$, which leads to $x = p \pm \sqrt{q}$. The distance between these two roots is:

$$|(p + \sqrt{q}) - (p - \sqrt{q})| = 2\sqrt{q}$$

Thus, the entire problem boils down to finding the value of $q$.

### The Symphony of the Arithmetic Progression

Now, let us introduce the four points $a_1, a_2, a_3, a_4$. We are told they are in an arithmetic progression with a mean of $p$. Since the mean is $p$, these points are perfectly balanced around the axis of symmetry $x = p$.

Instead of using the standard $a, a+d, a+2d, a+3d$, let us use a symmetric representation. Let the common difference be $2d$. We define our points as:

$$a_1 = p - 3d, \quad a_2 = p - d, \quad a_3 = p + d, \quad a_4 = p + 3d$$

When we plug these into $f(x) = (x-p)^2 - q$, the $p$ vanishes instantly. The calculations are as follows:

$$f(a_1) = f(p - 3d) = ((p - 3d) - p)^2 - q = 9d^2 - q$$

$$f(a_2) = f(p - d) = ((p - d) - p)^2 - q = d^2 - q$$

Due to the symmetry of the parabola, $f(a_4) = f(a_1)$ and $f(a_3) = f(a_2)$. We have reduced four complex points down to just two distinct values: $9d^2 - q$ and $d^2 - q$.

### The Algebraic Conflict

The problem states that $|f(a_i)| = 500$ for all $i$. This implies that the absolute value of our two distinct expressions must be $500$. Therefore, we have:

$$|9d^2 - q| = 500 \quad 	ext{and} \quad |d^2 - q| = 500$$

Since $d > 0$, it is undeniable that $9d^2 - q > d^2 - q$. They cannot both be $500$, and they cannot both be $-500$. The only logical conclusion is that the larger value must be positive, and the smaller value must be negative.

Thus, we lock in our system of equations:

$$9d^2 - q = 500$$

$$d^2 - q = -500$$

### The Final Resolution

We now solve the system of two linear equations in terms of $d^2$ and $q$. Subtracting the second equation from the first, we get:

$$(9d^2 - q) - (d^2 - q) = 500 - (-500)$$

$$8d^2 = 1000 \implies d^2 = 125$$

Substituting $d^2 = 125$ back into the second equation:

$$125 - q = -500 \implies q = 625$$

We know that the absolute difference between the roots is $2\sqrt{q}$. Substituting our value of $q$:

$$2\sqrt{625} = 2 	imes 25 = 50$$

The journey is complete. The final answer is **50**. Remember, in JEE Advanced, the math is rarely about brute force; it is about finding the path of least resistance through symmetry and logic.

Enter Numerical Value:

Visualized Solution

Analyze the Function $f (x)$

Roots of $f (x) = 0$

Target: Difference of Roots

Define the Arithmetic Progression

Evaluate $f (a_{i})$

Apply the Condition $∣ f (a_{i}) ∣ = 500$

Deduce the Signs

Set Up the Equations

Solve for $d^{2}$

Solve for $q$

Final Calculation

The Sigma Insight: Maximum and Minimum Values of Quadratic Expressions

Analyzing the Geometry of the Parabola

The Symphony of the Arithmetic Progression

The Algebraic Conflict

The Final Resolution

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Enter Numerical Value:

Visualized Solution

Analyze the Function .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(x)

Roots of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(x)=0

Target: Difference of Roots

Define the Arithmetic Progression

Evaluate .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } f(ai​)

Apply the Condition .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ∣f(ai​)∣=500

Deduce the Signs

Set Up the Equations

Solve for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } d2

Solve for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } q