Question

Find the ratio in which the point $\left( {\dfrac{{5 + a}}{7},\dfrac{{6a + 3}}{7}} \right)$ divides the join of $\left( {1,3} \right)$ and $\left( {2,7} \right)$. Also find the value of $a$.
A) $3:4, - 5$
B) $4:3, - 5$
C) $3:4, - 5$
D) Exists and is equal to $2$

Answer 1

Given two points and another point that divides the join of the two points. That means the point which divides the joint in some ratio, lies on the line joining the two points. Therefore, that point will satisfy the equation of the line. Using this we will find the value of $a$ . Then using the division formula, we will find the ratio.

Complete step by step solution:
It is given that the point $\left( {\dfrac{{5 + a}}{7},\dfrac{{6a + 3}}{7}} \right)$ divides the join of $\left( {1,3} \right)$ and $\left( {2,7} \right)$.
We will use the two-point form of the equation of a straight line and find the equation of line.
If a line passes through two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ then the equation of the line using two-point form is given by:
$\dfrac{{{y_2} - {y_1}}}{{y - {y_1}}} = \dfrac{{{x_2} - {x_1}}}{{x - {x_1}}}$
In the given problem the line passes through the points $\left( {1,3} \right)$ and $\left( {2,7} \right)$.
Therefore, the equation of the line will be:
$\dfrac{{7 - 3}}{{y - 3}} = \dfrac{{2 - 1}}{{x - 1}}$
On simplification we write:
$4\left( {x - 1} \right) = \left( {y - 3} \right)$
Therefore, the equation of the line is $4x - y = 1$ . … (1)
Since the point $\left( {\dfrac{{5 + a}}{7},\dfrac{{6a + 3}}{7}} \right)$ divides the join of the given points, it must also lie on the same line.
Therefore, $\left( {\dfrac{{5 + a}}{7},\dfrac{{6a + 3}}{7}} \right)$ satisfies the equation (1).
Substituting the respective values of the co-ordinates we write:
$4\left( {\dfrac{{5 + a}}{7}} \right) - \left( {\dfrac{{6a + 3}}{7}} \right) = 1$
This implies:
$20 + 4a - 6a - 3 = 7$
Solving for $a$, we get $a = 5$ .
Therefore, the point that divides the join of $\left( {1,3} \right)$ and $\left( {2,7} \right)$ is $\left( {\dfrac{{10}}{7},\dfrac{{33}}{7}} \right)$ .
Now if a point $(x,y)$ divides the join of the points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ in the ratio $m:n$ then the division formula states that :
$x = \dfrac{{m{x_2} + m{x_1}}}{{m + n}},y = \dfrac{{m{y_2} + m{y_1}}}{{m + n}}$
Let us assume that the point $\left( {\dfrac{{10}}{7},\dfrac{{33}}{7}} \right)$, divides the join of $\left( {1,3} \right)$ and $\left( {2,7} \right)$ in the ratio $m:n$ .
Using the division formula, we write:
$\dfrac{{10}}{7} = \dfrac{{2m + n}}{{m + n}},\dfrac{{33}}{7} = \dfrac{{7m + 3n}}{{m + n}}$
Now using the above two equalities we will form an equation in terms of $m$ and $n$.
The equation is given as follows:
$3n = 4m$
Therefore, $\dfrac{m}{n} = \dfrac{3}{4}$ .
Hence the given point divides the join of the points $\left( {1,3} \right)$ and $\left( {2,7} \right)$ in the ratio $3:4$ and the value of $a = 5$ .

Therefore, the correct option is D.

Note: First of all, it is important to note that the point of division lies on the same line that joins the two points that are given. This observation is important to find the value of the unknown otherwise it is impossible to find the point. Next important thing is we don’t have to find the exact value of $m$ and $n$ . Instead we just expressed one in terms of the other so that we could find the ratio.