The line x+y=4 cuts the line joining P(-1,1) and Q(5,7) at R. What is PR: RQ equal to?

Explanation

Let the ratio be m:n. By the section formula, the coordinates of R are (\frac{5m-n}{m+n}, \frac{7m+n}{m+n}). Since R lies on the line x+y=4, substituting the coordinates yields \frac{5m-n}{m+n} + \frac{7m+n}{m+n} = 4. This simplifies to \frac{12m}{m+n} = 4 \implies 12m = 4m+4n \implies 8m = 4n \implies \frac{m}{n} = \frac{1}{2}. The ratio is 1:2.

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