question_answer

The point (-4, 6) divides the line segment joining the points \[A\left( -6,\text{ }10 \right)\]and B (3,-8). The ratio is

A) 1 : 2

B) 7 : 2

C) 2 : 7

D) 4 : 1

Correct Answer: C

Solution :

Let point \[P\left( -4,\,6 \right)\]divides the line segment joining the points A (- 6,10) and B (3, - 8) in the ratio \[{{m}_{1}}:\,\,{{m}_{2}}\].

By using section formula, we get

\[\left( -4,\,6 \right)=\left( \frac{3{{m}_{1}}-6{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}},\frac{-8{{m}_{1}}+10{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}} \right)\]                                                                                                                                      …. (i)

On equating .v-coordinate from both sides of Eq. (i), we get

\[-4=\frac{3{{m}_{1}}-6{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}}\]

\[\Rightarrow \,\,-4\left( {{m}_{1}}+{{m}_{2}} \right)=3{{m}_{1}}-6{{m}_{2}}\]

\[\Rightarrow \,\,\,-4{{m}_{1}}-4{{m}_{2}}=3{{m}_{1}}-6{{m}_{2}}\]

\[\Rightarrow \,-4\,{{m}_{1}}-3{{m}_{1}}=-6{{m}_{2}}+4{{m}_{2}}\]

\[\Rightarrow \,\,\,\,\,-7\,{{m}_{1}}=-2{{m}_{2}}\]

\[\Rightarrow \,\,\frac{{{m}_{1}}}{{{m}_{2}}}=\frac{2}{7}\]

\[{{m}_{1}}\,:\,{{m}_{2}}=2:7\]