Animated Solution for Mathematics - Straight Lines: Consider the set of all lines $px+qy+r=0$ such that $3p+2q+4r=0$. Which one of the following statements is true ?

Imagine you are standing on a vast coordinate plane. You have a line defined by the equation $px+qy+r=0$. Here, $p$, $q$, and $r$ are the architects of the line's position and orientation.

However, these coefficients are bound by a specific constraint: $3p+2q+4r=0$. This is not merely an algebraic nuisance; it is a geometric mandate that forces the coefficients to be dependent on one another.

To uncover the hidden truth, we must align our perspective. Look at our line equation $px+qy+r=0$ and our constraint $3p+2q+4r=0$.

The line equation has a coefficient of $1$ for $r$, while our constraint has a coefficient of $4$ for $r$. To make them speak the same language, we must normalize the constraint by dividing the entire equation by $4$:

This simple act of division is the key that unlocks the door. It yields the following expression:

Now, place the two equations side by side. The general line equation is $p(x)+q(y)+r=0$, and our transformed constraint is:

By direct comparison, we can see that for any line in this family, $x$ must be $\frac{3}{4}$ and $y$ must be $\frac{1}{2}$.

This means that no matter how you vary $p$, $q$, and $r$, as long as they satisfy the original constraint, the line will always pass through the fixed point:

This is the definition of concurrency. All these lines, despite their different slopes, are anchored to this single, fixed point. You have just uncovered the secret of the family of lines, a powerful tool for your JEE journey.