Animated Solution for Mathematics - Straight Lines: The line $L$ given by $\frac{x}{5}+\frac{y}{b}=1$ passes through the point $(13,32)$. The line $K$ is parallel to $L$ and has the equation $\frac{x}{c}+\frac{y}{3}=1$. Then the distance between $L$ and $K$ is

Imagine you are standing on a vast, two-dimensional plane. You have two lines, $L$ and $K$, stretching out into infinity. They are perfectly parallel, never touching, yet they are defined by the subtle interplay of their intercepts.

We begin with line $L$, defined by the equation:

We are told it passes through the point $P(13,32)$. In the language of coordinate geometry, this means the point $P$ is a prisoner of this line; its coordinates must satisfy the equation.

By substituting $x=13$ and $y=32$, we get:

With a bit of algebraic finesse, we isolate the term:

Solving for $b$, we find $b=-20$. Now, our line $L$ is fully revealed:

To make this line easier to work with, we multiply by $20$ to reach the standard form:

Now, consider line $K$, defined by:

We are told $K$ is parallel to $L$. In the world of lines, parallelism is a strict condition: it means they must share the exact same slope.

From our standard form of $L$, $4x-y-20=0$, we can see the slope is $4$. For line $K$, the slope is $-\frac{3}{c}$. Setting these equal, we have:

Substituting this back into the equation for $K$, we get:

Multiplying by $3$, we transform this into $-4x+y=3$, or:

We have arrived at the climax of our journey. We have two parallel lines in their most powerful form:

The distance $d$ between two parallel lines $Ax+By+C_1=0$ and $Ax+By+C_2=0$ is given by the elegant formula:

Here, $A=4$, $B=-1$, $C_1=-20$, and $C_2=3$. Plugging these into our formula, we calculate:

This simplifies to:

And there it is—the exact distance between these two parallel paths. The final result is: