Animated Solution for Mathematics - Sequence and Series: If $x_1,x_2,x_3$ and $y_1,y_2,y_3$ are both in G.P. with the same common ratio, then the points $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$

Imagine you are standing on a coordinate plane, looking at three points floating in space. At first glance, they seem like random coordinates: $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$.

But there is a secret rhythm governing their placement. We are told that both the $x$-coordinates and the $y$-coordinates follow a Geometric Progression (G.P.) with the exact same common ratio, $r$.

This isn't just a coincidence; it is a mathematical symphony waiting to be decoded.

Let us strip away the mystery. If $x_1,x_2,x_3$ are in G.P., we can define them using a starting value $a$ and a common ratio $r$. Thus, $x_1=a$, $x_2=ar$, and $x_3=a{r}^2$.

Similarly, for the $y$-coordinates, we use a starting value $b$ and the same ratio $r$, giving us $y_1=b$, $y_2=br$, and $y_3=b{r}^2$.

Now, look at our points again. They aren't just random; they are parametric representations:

Do you see the elegance? Every point is just a scaled version of the previous one. We are essentially multiplying the coordinates by the same factor $r$ to jump from one point to the next.

To determine if these points lie on a straight line, we must check if the slope between them remains constant. If the slope between $P_1$ and $P_2$ is the same as the slope between $P_2$ and $P_3$, then the points must be collinear.

Let us calculate the slope $m_1$ between $P_1$ and $P_2$:

I know that looking at variables can be intimidating, but take a breath. Factor out the common terms. In the numerator, we have $b(r-1)$, and in the denominator, we have $a(r-1)$.

Assuming $r eq 1$ (because if $r=1$, the points are identical, which is a trivial case), we can cancel the $(r-1)$ term. We are left with:

Now, let us test the slope $m_2$ between $P_2$ and $P_3$:

Again, let us factorize. We can pull out $br$ from the numerator and $ar$ from the denominator:

Watch closely as the $r(r-1)$ terms vanish into thin air. What remains?

Since $m_1=m_2=\frac{b}{a}$, the slope is constant throughout the entire sequence of points. Geometrically, this means that every point $(x_n,y_n)$ lies on the line $y=\frac{b}{a}x$.

We have successfully proven that these points are not scattered randomly; they are perfectly aligned on a straight line passing through the origin.

This is the beauty of the JEE Advanced curriculum—it takes a seemingly abstract sequence problem and reveals the rigid, beautiful geometry hidden underneath. You didn't just solve a problem; you uncovered a fundamental truth about how sequences map onto the Cartesian plane. Keep this intuition with you; it will serve you well in your journey toward mastery.