In what ratio is the line segment joining the | vedclass

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(A) Let the line $x + y = 4$ divide the line segment joining $A(-1, 1)$ and $B(5, 7)$ in the ratio $k: 1$ at point $P(x, y)$.By the section formula,th

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$1: 2$

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(A) Let the line $x + y = 4$ divide the line segment joining $A(-1, 1)$ and $B(5, 7)$ in the ratio $k: 1$ at point $P(x, y)$.By the section formula,the coordinates of $P$ are:$P = \left( \frac{k(5) + 1(-1)}{k + 1}, \frac{k(7) + 1(1)}{k + 1} \right) = \left( \frac{5k - 1}{k + 1}, \frac{7k + 1}{k + 1} \right)$Since $P$ lies on the line $x + y = 4$,we substitute these coordinates into the equation:$\frac{5k - 1}{k + 1} + \frac{7k + 1}{k + 1} = 4$$\frac{5k - 1 + 7k + 1}{k + 1} = 4$$\frac{12k}{k + 1} = 4$$12k = 4(k + 1)$$12k = 4k + 4$$8k = 4$$k = \frac{4}{8} = \frac{1}{2}$Thus,the ratio $k: 1$ is $1: 2$.

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

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