Animated Solution for Mathematics - Straight Lines: The foot of the perpendicular drawn from the origin, on the line, $3x+y=\lambda (\lambda \ne 0)$ is P. If the line meets x-axis at A and y-axis at B, then the ratio BP : PA is :-

Welcome, fellow traveler on the path to JEE mastery. Today, we are not just solving a problem; we are uncovering the hidden elegance of coordinate geometry.

Imagine you are standing on a Cartesian plane. You have a line, defined by the equation $3x+y=\lambda$. It cuts through the axes, creating a segment between the x-axis and the y-axis.

We are going to drop a perpendicular from the origin $O(0,0)$ to this line, hitting it at a point $P$. Our mission is to find the ratio in which this point $P$ divides the segment $BA$.

Defining the Boundaries

Before we dive into the complex algebra, let's ground ourselves. We need to find the intercepts of the line.

For the x-intercept $A$, we set $y=0$. The equation $3x+0=\lambda$ gives us $x=\frac{\lambda }{3}$. So, our point $A$ is at $(\frac{\lambda }{3},0)$.

Similarly, for the y-intercept $B$, we set $x=0$. The equation $3(0)+y=\lambda$ gives us $y=\lambda$. Thus, point $B$ is at $(0,\lambda )$. We now have the two endpoints of our segment, $A$ and $B$.

The Perpendicular Descent

Now, we drop the perpendicular from the origin $O(0,0)$ to the line $3x+y-\lambda =0$. Let the foot of this perpendicular be $P(h,k)$.

We utilize the foot-of-perpendicular formula. For a line $ax+by+c=0$ and a point $(x_1,y_1)$, the foot $(h,k)$ satisfies:

Substituting our values, where $(x_1,y_1)=(0,0)$, $a=3$, $b=1$, and $c=-\lambda$, we get:

Simplifying this, we find:

This leads us directly to $h=\frac{3\lambda }{10}$ and $k=\frac{\lambda }{10}$. Our point $P$ is $(\frac{3\lambda }{10},\frac{\lambda }{10})$.

The Ratio of Elegance

We need the ratio $BP:PA$. Let this ratio be $m:n$. Using the section formula for the x-coordinate, we have:

Substituting our values:

Since $\lambda eq 0$, we can cancel it out, leaving:

Cross-multiplying gives $9(m+n)=10m$, which simplifies to $9m+9n=10m$, or $m=9n$. Thus, the ratio $m:n$ is $9:1$.

As you grow in your JEE preparation, you will start to see these patterns everywhere. For any line $ax+by=c$, the ratio in which the foot of the perpendicular from the origin divides the segment between the axes is always $a^2:b^2$.

Here, $3^2:1^2=9:1$. It is a beautiful, symmetric result. Keep practicing, keep visualizing, and remember: the math is not just about the answer; it is about the journey.