Animated Solution for Mathematics - Differentiation: If the lines $x+y=a$ and $x-y=b$ touch the curve $y=x^2-3x+2$ at the points where the curve intersects the $x$-axis, then $\frac{a}{b}$ is equal to

A Journey Through Tangents

Imagine you are standing on a coordinate plane, looking at the graceful curve of a parabola defined by the equation $y=x^2-3x+2$. It is a classic shape, opening upwards, sweeping down to cross the $x$-axis and then rising again.

Our mission today is to find the two lines that touch this curve exactly where it kisses the $x$-axis. This is not just a calculation; it is a story of how calculus reveals the hidden geometry of curves.

Finding the Roots

Before we can talk about tangents, we must find the points of contact. The problem tells us these tangents exist where the curve intersects the $x$-axis.

On the $x$-axis, the vertical position is always zero. So, we set our equation to zero:

This is a simple quadratic equation. We look for two numbers that multiply to $2$ and add to $-3$. Those numbers are $-1$ and $-2$.

Thus, we factorize the expression as $(x-1)(x-2)=0$. The roots are $x=1$ and $x=2$. Our points of contact are $A(1,0)$ and $B(2,0)$. We have successfully anchored our geometry!

The Derivative's Magic

Now, we need the slopes of the tangents at these points. This is where calculus shines.

The derivative of a function tells us the slope of the tangent at any point. Let us differentiate $y=x^2-3x+2$ with respect to $x$:

This formula, $\frac{dy}{dx}=2x-3$, is our slope generator. It tells us exactly how steep the curve is at any $x$-coordinate we choose. It is the heartbeat of the parabola.

Constructing the Tangents

Let us find the slope at point $A(1,0)$. We substitute $x=1$ into our derivative:

With a slope of $-1$ and a point $(1,0)$, we use the point-slope form $y-y_1=m(x-x_1)$:

This matches the form $x+y=a$, so we immediately see that $a=1$. Now, for point $B(2,0)$, we substitute $x=2$ into the derivative:

With a slope of $1$ and a point $(2,0)$, we again use the point-slope form:

This matches the form $x-y=b$, so we find that $b=2$.

The Final Comparison

We have arrived at the finish line. We found $a=1$ and $b=2$. The problem asks for the ratio $\frac{a}{b}$.

Substituting our values, we get:

And there it is! A beautiful, clean result. We started with a simple parabola, used the power of derivatives to find the slopes, and constructed the tangent lines to reveal the constants $a$ and $b$.

The final result is $\frac{1}{2}$. Mathematics is not just about numbers; it is about uncovering the elegant relationships hidden within the curves.