Solve this


If the point $P\left(\frac{1}{2}, y\right)$ lies on the line segment joining the points $A(3,-5)$ and $B(-7,9)$ then find the

ratio in which P divides AB. Also, find the value of y.



Let the point $P\left(\frac{1}{2}, y\right)$ divides the line segment joining the points $A(3,-5)$ and $B(-7,9)$ in the ratio $k: 1$. Then

$\left(\frac{1}{2}, y\right)=\left(\frac{k(-7)+3}{k+1}, \frac{k(9)-5}{k+1}\right)$

$\Rightarrow \frac{-7 k+3}{k+1}=\frac{1}{2}$ and $\frac{9 k-5}{k+1}=y$

$\Rightarrow k+1=-14 k+6 \Rightarrow k=\frac{1}{3}$

Now, substituting $k=\frac{1}{3}$ in $\frac{9 k-5}{k+1}=y$, we get

$\frac{\frac{9}{3}-5}{\frac{1}{3}+1}=y \Rightarrow y=\frac{9-15}{1+3}=-\frac{3}{2}$

Hence, required ratio is $1: 3$ and $y=-\frac{3}{2}$.


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