In what ratio is the line segment joining the points $(3, -5)$ and $(-1, 6)$ divided by the line $y = x$?

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Step 1: Let the required ratio be $K : 1$. Coordinates of $P$ are $\left(\frac{-K + 3}{K + 1}, \frac{6K - 5}{K + 1}\right)$
Step 2: Point $P$ lies on $y = x$, so $\frac{6K - 5}{K + 1} = \frac{-K + 3}{K + 1}$
Step 3: $6K - 5 = -K + 3 \Rightarrow 7K = 8 \Rightarrow K = \frac{8}{7}$
Step 4: Required ratio is $8 : 7$

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