Find the ratio in which the point P left(frac | vedclass

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Question

(A) Let the point $P \left(\frac{3}{4}, \frac{5}{12}\right)$ divide the line segment joining $A \left(\frac{1}{2}, \frac{3}{2}\right)$ and $B(2, -5)$

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(A) Let the point $P \left(\frac{3}{4}, \frac{5}{12}ight)$ divide the line segment joining $A \left(\frac{1}{2}, \frac{3}{2}ight)$ and $B(2, -5)$ in the ratio $k: 1$.Using the section formula,the coordinates of $P$ are given by:$P = \left(\frac{k(2) + 1(\frac{1}{2})}{k+1}, \frac{k(-5) + 1(\frac{3}{2})}{k+1}ight)$Equating the $x$-coordinate:$\frac{3}{4} = \frac{2k + \frac{1}{2}}{k+1}$$3(k+1) = 4(2k + \frac{1}{2})$$3k + 3 = 8k + 2$$3 - 2 = 8k - 3k$$1 = 5k$$k = \frac{1}{5}$Thus,the ratio $k: 1$ is $\frac{1}{5}: 1$,which is $1: 5$.Checking with the $y$-coordinate:$y = \frac{-5(\frac{1}{5}) + \frac{3}{2}}{\frac{1}{5} + 1} = \frac{-1 + \frac{3}{2}}{\frac{6}{5}} = \frac{\frac{1}{2}}{\frac{6}{5}} = \frac{1}{2} 	imes \frac{5}{6} = \frac{5}{12}$.Since this matches the given $y$-coordinate,the ratio is $1: 5$.

Find the ratio in which the point $P \left(\frac{3}{4}, \frac{5}{12}\right)$ divides the line segment joining the points $A \left(\frac{1}{2}, \frac{3}{2}\right)$ and $B(2, -5)$. (in $: 5$)

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