If $A$ and $B$ are $(1, 4)$ and $(5, 2)$ respectively, find the coordinates of $P$ when $\frac{AP}{BP} = \frac{3}{4}$.
Given:
$A$ and $B$ are $(1, 4)$ and $(5, 2)$ respectively.
To do:
We have to find the coordinates of $P$ when $\frac{AP}{BP} = \frac{3}{4}$.
Solution:
Let the coordinates of $P$ be $(x,y)$.
Point $P$ divides the line segment joining the points $A(1, 4)$ and $B(5, 2)$ in the ratio of $3 : 4$.
Using section formula, we have,
\( (x, y)=(\frac{mx_{2}+nx_{1}}{m+n}, \frac{my_{2}+ny_{1}}{m+n}) \)
Therefore,
\( P(x,y)=\left(\frac{3 \times 5+4 \times 1}{3+4}, \frac{3 \times 2+4 \times 4}{3+4}\right) \)
\( =\left(\frac{15+4}{7}, \frac{6+16}{7}\right) \)
\( =\left(\frac{19}{7}, \frac{22}{7}\right) \)
Therefore, the coordinates of $P$ are $(\frac{19}{7}, \frac{22}{7})$.
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