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

Question:

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

Solution:

Let the required ratio be $k: 1$.

Then, by section formula, the coordinates of C are

$C\left(\frac{7 k+2}{k+1}, \frac{8 k+3}{k+1}\right)$

Therefore,

$\frac{7 k+2}{k+1}=4$ and $\frac{8 k+3}{k+1}=5 \quad[\because C(4,5)$ is given $]$

$\Rightarrow 7 k+2=4 k+4$ and $8 k+3=5 k+5 \Rightarrow 3 k=2$ and $3 k=2$

$\Rightarrow k=\frac{2}{3}$ in each case

So, the required ratio is $\frac{2}{3}: 1$, which is same as $2: 3$.

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