Question

In what ratio does the point C(4, 5) divide the join of A(2, 3) and B(7, 8)?

Answer 1

Let the required ratio be $k:1$.
Then, by section formula, the coordinates of C are
$C\left(\frac{7k+2}{k+1},\frac{8k+3}{k+1}\right)$
Therefore,
$$\begin{gathered} \frac{7k+2}{k+1}=4\text{and}\frac{8k+3}{k+1}=5\left[\because C\left(4,5\right)\mathrm{is}\mathrm{given}\right] \\ \Rightarrow 7k+2=4k+4\text{and}8k+3=5k+5\Rightarrow 3k=2\text{and}3k=2 \end{gathered}$$
$\Rightarrow k=\frac{2}{3}\text{in each case}$
$\text{So, the required ratio is}\frac{2}{3}:1,\text{which is same as 2:3.}$