Animated Solution for Mathematics - Straight Lines: Three lines $px+qy+r=0,qx+ry+p=0$ and $rx+py+q=0$ are concurrent if

In the study of coordinate geometry, three lines are said to be concurrent if they intersect at a single, common point. For three lines given by the equations:

The condition for concurrency is that the determinant of their coefficients must vanish.

We utilize the determinant of the coefficient matrix to establish the condition for concurrency. Mathematically, this is expressed as:

This determinant represents the geometric constraint required for the three lines to share a common point of intersection.

Extracting the coefficients from the given lines, we observe a distinct cyclic symmetry: $(p,q,r)$, $(q,r,p)$, and $(r,p,q)$. Substituting these into our determinant, we expand along the first row:

Distributing the terms, we obtain:

Simplifying this expression leads us to the fundamental result:

To explore the implications of this result, we recall the classic algebraic identity:

Since we have established that $p^3+q^3+r^3-3pqr=0$, the product of these two factors must equal zero. This implies that at least one of the factors must be zero.

We identify two primary conditions for concurrency:

1. $p+q+r=0$ 2. $p^2+q^2+r^2-pq-qr-rp=0$, which can be rearranged as $p^2+q^2+r^2=pq+qr+rp$.

These conditions are mathematically equivalent expressions of the same geometric reality. The concurrency of the lines is guaranteed if either of these algebraic constraints is satisfied.