In what ratio does the line y - x + 2 = 0 div | vedclass

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(C) Let the line $y - x + 2 = 0$ divide the line segment joining $A(3, -1)$ and $B(8, 9)$ in the ratio $\lambda : 1$.Using the section formula,the coo

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$2 : 3$

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(C) Let the line $y - x + 2 = 0$ divide the line segment joining $A(3, -1)$ and $B(8, 9)$ in the ratio $\lambda : 1$.Using the section formula,the coordinates of the point of division are $\left( \frac{8\lambda + 3}{\lambda + 1}, \frac{9\lambda - 1}{\lambda + 1} \right)$.Since this point lies on the line $y - x + 2 = 0$,we substitute the coordinates into the equation:$\frac{9\lambda - 1}{\lambda + 1} - \frac{8\lambda + 3}{\lambda + 1} + 2 = 0$Multiplying by $(\lambda + 1)$,we get:$(9\lambda - 1) - (8\lambda + 3) + 2(\lambda + 1) = 0$$9\lambda - 1 - 8\lambda - 3 + 2\lambda + 2 = 0$$3\lambda - 2 = 0$$\lambda = \frac{2}{3}$Thus,the required ratio is $2 : 3$.

In what ratio does the line $y - x + 2 = 0$ divide the line segment joining the points $(3, -1)$ and $(8, 9)$?

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