Q.In what ratio, the line joining $(-1, 1)$ and $(5, 7)$ is divided by the line $x + y = 4$?

Concept: Section Formula – we find the point where the line $x+y=4$ cuts the segment, then use the ratio formula.

Step 1: Let the required ratio be $k:1$ (internal division). The coordinates of the point dividing $(-1,1)$ and $(5,7)$ in ratio $k:1$ are

Step 2: This point lies on $x+y=4$. Substitute:

Step 3: Simplify numerator: $(5k-1)+(7k+1)=12k$. So …

We need the ratio in which the line $x + y = 4$ cuts the segment joining $A(-1, 1)$ and $B(5, 7)$. The dividing line is not a point — it's a whole line. So the intersection point $P$ of $x + y = 4$ with $AB$ is the actual point of division. Once we find $P$, we use the section formula to get the ratio.

The core idea: Section Formula

If a point $P(x, y)$ divides the segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in the ratio $m : n$ (i.e., $AP : PB = m : n$), then:

We don't know $m$ and $n$ yet. But we do know that $P$ lies on $x + y = 4$. So we can set up an equation.

Step-by-step solution

1. Let the ratio be $k : 1$

Assume $P$ divides $AB$ internally in the ratio $k : 1$, meaning $AP : PB = k : 1$. Then using the section formula with $A(-1, 1)$ and $B(5, 7)$:

So:

2. Use the condition that $P$ lies on $x + y = 4$

Substitute $x$ and $y$ into the line equation:

Since denominators are the same, combine numerators:

$\frac{(5k - 1) + (7k + 1)}{k + 1} = 4$ …