question_answer

The coordinates of the point which divides the line segment joining the points (4, - 3) and (9, 7) internally in the ratio 3: 2 is

A) (7, 3)

B) (3, 7)

C) (35, 15)

D) (27, 21)

Correct Answer: A

Solution :

Let P (x, y) be the required point. Then, P divides AB internally in die ratio 3 : 2.

Here, \[\frac{{{m}_{1}}}{{{m}_{2}}}=\frac{3}{2}\] and \[\left( {{x}_{1}},\,{{y}_{1}} \right)=\left( 4,\,-3 \right)\]\[\left( {{x}_{2}},\,{{y}_{2}} \right)=\left( 9,\,7 \right)\]

Then, \[P\left( x,\,y \right)=\]

\[P\left( \frac{{{m}_{1}}{{x}_{2}}+{{m}_{2}}{{x}_{1}}}{{{m}_{1}}+{{m}_{2}}},\,\frac{{{m}_{1}}{{y}_{2}}+{{m}_{2}}{{y}_{1}}}{{{m}_{1}}+{{m}_{2}}} \right)\]

[by section formula]

\[=P\left( \frac{3\times 9+2\times 4}{3+2},\,\frac{3\times 7+2\times \left( -3 \right)}{3+2} \right)\]

\[=P\left( \frac{27+8}{5},\frac{21-6}{5} \right)=P\left( \frac{35}{5},\,\frac{15}{5} \right)\]

=P (7,3)

Therefore, (7, 3) is the required point.