Question

Determine the ratio in which the line $2x + y - 4 = 0$ divides the line segment joining the points $A(2, - 2)$ and $B(3,7)$.

Answer 1

Hint: First we have to assume that the line divides AB in the ratio of $k:1$ at the point $P$
Then we have to find the coordinates of $P$ by using the section formula.
We will substitute the coordinates of it in the given line.
Finally we will get the required result.

Complete step-by-step answer:
It is given that the line segment joining the points $A(2, - 2)$ and $B(3,7)$
Also given that equation of a line $2x + y - 4 = 0$
We have to assume that the line $2x + y - 4 = 0$ divides \[\;{\text{AB}}\] in the ratio $k:1$ at the point \[{\text{P}}\]
Here the diagram as follows,

If $P$ divides the line segment joining $A$ and $B$ in the ratio\[{\text{m:n}}\], then we have to use the section formula.
Here is the section formula $\left( {\dfrac{{{x_2}m + {x_1}n}}{{n + m}},\dfrac{{{y_2}m + {y_1}n}}{{n + m}}} \right)$
Since the vertices of the $A(2, - 2)$ and $B(3,7)$
So we can write it as,
$\left( {{x_1},{y_1}} \right) = \left( {2, - 2} \right)$
$\left( {{x_2},{y_2}} \right) = \left( {3,7} \right)$
Putting the value section formula we get,
Coordinates of point ${\text{P = }}\left[ {\dfrac{{k(3) + 1(2)}}{{k + 1}},\dfrac{{k(7) + 1( - 2)}}{{k + 1}}} \right]$
On some simplification we get,
 ${\text{P}}\left( {x,y} \right) = \left[ {\dfrac{{3k + 2}}{{k + 1}},\dfrac{{7k - 2}}{{k + 1}}} \right]$
Since we have point \[{\text{P}}\] on the line $2x + y - 4 = 0$
We will put the points, $x = \dfrac{{3k + 2}}{{k + 1}}$ and $y = \dfrac{{7k - 2}}{{k + 1}}$ in the given equation of line.
That is, $2x + y - 4 = 0$
$2\left( {\dfrac{{3k + 2}}{{k + 1}}} \right) + \left( {\dfrac{{7k + 2}}{{k + 1}}} \right) - 4 = 0$
Take the LCM, we get
$\dfrac{{2(3k + 2) + (7k - 2) - 4(k + 1)}}{{(k + 1)}} = 0$
Let us multiply the numerator terms and the denominator should be in RHS, which means that we take a cross multiplication we get,
$6k + 4 + 7k - 2 - 4k - 4 = 0(k + 1)$
On adding some terms we get,
$9k - 2 = 0$
Equate the terms we get,
$9k = 2$
Let us divided it,
$k = \dfrac{2}{9}$
Thus the ratio in which \[{\text{P}}\] divides \[{\text{AB}}\] is $k:1$
Putting the value of k we get,
 = $\dfrac{2}{9}:1$
Let us take a cross multiply the ratio terms we get,
 =$2:9$
Hence the ratio is $2:9$
$\therefore $ The required ratio in which the line $2x + y - 4 = 0$ divides the line segment joining the points $A$ and $B$ is $2:9$.

Note: The possibility for mistake here is to writing the section formula for points $A({x_1},{y_1})$ and $B({x_2},{y_2})$
Wrongly as $x = \dfrac{{m{x_1} + n{x_2}}}{{m + n}}$ , $y = \dfrac{{m{y_1} + n{y_2}}}{{n + m}}$
Instead of $x = \dfrac{{m{x_2} + n{x_1}}}{{m + n}}$ ,$y = \dfrac{{m{y_2} + n{y_1}}}{{m + n}}$
It is impossible to find three variables from two equations.
We need only two equations to find two variables.
And the ratio between \[{\text{m}}\] and \[{\text{n}}\].