Animated Solution for Mathematics - Differentiation: If the tangent to the curve $y=x^3-x^2+x$ at the point $(a,b)$ is also tangent to the curve $y=5x^2+2x-25$ at the point $(2,-1)$, then $|2a+9b|$ is equal to ________

Welcome, future engineer. Today, we embark on a journey through the elegant world of calculus. We are not just solving a problem; we are uncovering the hidden geometry that connects two seemingly unrelated curves.

Imagine you are standing on a graph, looking at two distinct paths. One is a cubic curve, $y=x^3-x^2+x$, and the other is a parabola, $y=5x^2+2x-25$. The problem states that a single, straight line acts as a bridge, grazing both of these curves as a common tangent.

The Parabola (The Anchor)

We start with the parabola because we have a solid anchor point: $(2,-1)$. To find the slope of the tangent line here, we calculate the derivative:

At $x=2$, the slope $m$ is $10(2)+2=22$. We have now locked down the steepness of our line.

Using the point-slope form $y-y_1=m(x-x_1)$, we substitute our values: $y-(-1)=22(x-2)$. Expanding this, we obtain the equation of our common tangent line:

The Cubic Curve (The Challenge)

Now, we shift our focus to the cubic curve $y=x^3-x^2+x$. We know this same line is tangent to the cubic curve at some unknown point $(a,b)$.

If we differentiate the cubic equation, we get:

At the point where $x=a$, this derivative must perfectly match the slope of our tangent line, which we know is $22$. Setting these equal, we get $3a^2-2a+1=22$, which simplifies to the quadratic equation:

The Trap of the Extraneous Root

Solving $3a^2-2a-21=0$ by splitting the middle term, we get $(3a+7)(a-3)=0$. This yields two candidates: $a=3$ and $a=-\frac{7}{3}$.

For $a=3$, the $y$-coordinate is $b=3^3-3^2+3=21$. We check if the point $(3,21)$ lies on our tangent line $y=22x-45$:

It matches perfectly. Now, for $a=-\frac{7}{3}$, the $y$-coordinate is $b=-\frac{553}{27}$. Checking this on the line, we find it does not satisfy the equation, meaning the tangent at $a=-\frac{7}{3}$ is parallel to our line but is not the same line. We reject this case.

With $a=3$ and $b=21$ confirmed, the final step is a simple calculation:

We have arrived at the final answer, 195. This problem demonstrates that in the JEE, the math is only half the battle; the other half is the geometric intuition and the rigor to verify every step.