Find the ratio in which the point P(m, 6) divides the join of A(−4, 3) and B(2, 8). Also, find the value of m.


Find the ratio in which the point P(m, 6) divides the join of A(−4, 3) and B(2, 8). Also, find the value of m.


Let the point P(m, 6) divide the line AB in the ratio k : 1.
Then, by the section formula:

$x=\frac{m x_{2}+n x_{1}}{m+n}, y=\frac{m y_{2}+n y_{1}}{m+n}$

The coordinates of $P$ are $(m, 6)$.

$m=\frac{2 k-4}{k+1}, 6=\frac{8 k+3}{k+1}$

$\Rightarrow m(k+1)=2 k-4,6 k+6=8 k+3$

$\Rightarrow m(k+1)=2 k-4,6-3=8 k-6 k$

$\Rightarrow m(k+1)=2 k-4,2 k=3$

$\Rightarrow m(k+1)=2 k-4, k=\frac{3}{2}$

Therefore, the point $P$ divides the line $A B$ in the ratio $3: 2$.

Now, put ting the value of $k$ in the equation $m(k+1)=2 k-4$, we get:


$\Rightarrow m\left(\frac{3+2}{2}\right)=3-4$

$\Rightarrow \frac{5 m}{2}=-1 \Rightarrow 5 m=-2 \Rightarrow m=-\frac{2}{5}$

Therefore, the value of $m=-\frac{2}{5}$

So, the coordinates of Pare $\left(-\frac{2}{5}, 6\right)$.

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