Question

If $2a + 3b - 5c = 0$ , then the ratio in which $c$ divides $AB$ is ?
A) 3:2 internally
B) 3:2 externally
C) 2:3 internally
D) 2:3 externally

Answer 1

Hint: In order to solve this question, first we have to identify that this question is from linear algebra.
So the concept used in this question is from linear algebra. To solve this question you must have to know about the section formulas for internal division and external division and then compare the final term with that formal and you will get your answer.

Complete step by step answer:
First of all let’s see the section formula so you can remember what the section formula is.
Let us consider that the line segment connecting P and Q is divided by a point R lying on PQ. The point R can divide the line segment PQ in two ways: internally and externally.
Let's see both these cases individually.
Case 1:
Let us consider that the point R divides the line segment PQ in the ratio m: n in between the line PQ, given that m and n are positive scalar quantities we can write formula for internal division,
$ \Rightarrow \overrightarrow r = \dfrac{{m\overrightarrow b + n\overrightarrow a }}{{m + n}}$
Case 2:
Let us consider that the point R divides the line segment PQ in the ratio m: n, outside the line PQ, given that m and n are positive scalar quantities we can write formula for internal division,
$ \Rightarrow \overrightarrow r = \dfrac{{m\overrightarrow b - n\overrightarrow a }}{{m - n}}$
Now, let’s see our equation,
$ \Rightarrow 2a + 3b - 5c = 0$
We have to find that $c$ divides $AB$ in which ration so we take $c$ in right side,
$ \Rightarrow 2a + 3b = 5c$
Now, make divide by $5$ on both the side,
$ \Rightarrow \dfrac{{2a + 3b}}{5} = c$
Now, write $5$ in terms of $2$ and $3$ ,
$ \Rightarrow \dfrac{{2a + 3b}}{{2 + 3}} = c$
We can clearly see that this formula looks familiar to the internal division.
Here, $m = 2$ and $n = 3$ so we can say that,
$c$ divides $AB$ in 2:3 internally.
Therefore, the correct answer is option (C).

Note:
In coordinate geometry, the Section formula is used to find the ratio in which a line segment is divided by a point internally or externally. It is used to find out the centroid, incenters and excenters of a triangle. It is also use in physics, it is used to find the center of mass of systems, equilibrium points, etc. there is also section formula for midpoint,
$ \Rightarrow \overrightarrow r = (\dfrac{{{x_1} + {y_1}}}{2} + \dfrac{{{x_2} + {y_2}}}{2})$