A(k, 2) and B(3, 5) are the given points. If | vedclass

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(C) Let the point $P(t, t)$ divide the line segment joining $A(k, 2)$ and $B(3, 5)$ in the ratio $m:n = k:1$.Using the section formula, the coordinate

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$-2$

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(C) Let the point $P(t, t)$ divide the line segment joining $A(k, 2)$ and $B(3, 5)$ in the ratio $m:n = k:1$.Using the section formula, the coordinates of $P$ are given by:$P(x, y) = \left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} ight)$Substituting the values:$t = \frac{k(3) + 1(k)}{k + 1} = \frac{4k}{k + 1}$ --- $(1)$$t = \frac{k(5) + 1(2)}{k + 1} = \frac{5k + 2}{k + 1}$ --- $(2)$Since both expressions equal $t$, we equate them:$\frac{4k}{k + 1} = \frac{5k + 2}{k + 1}$Assuming $k 
eq -1$, we get:$4k = 5k + 2$$4k - 5k = 2$$-k = 2$$k = -2$

$A(k, 2)$ and $B(3, 5)$ are the given points. If the point $P(t, t)$ divides the line segment $\overline{AB}$ in the ratio $k:1$ starting from $A$, find the value of $k$.

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