Question

Find the ratio in which the line segment joining A (1, – 5) and B (– 4, 5) is divided by the x-axis. Also, find the coordinates of the point of division.

Answer 1

Hint: In order to solve this question, we should have some knowledge of section formula, that is when two points \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] are divided by any point (h, k) in the ratio m:n, then we can say, \[h=\dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n}\] and \[k=\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n}\].

Complete step-by-step answer:
In this question, we have to find the coordinates of a point that lies on the x-axis and divide the line joining points A (1, – 5) and B (– 4, 5) and also the ratio in which the point divides the line. Let the ratio in which the point divides the line be m:1. And we know that when a point lies on the x-axis then its y – coordinate is 0, which means the coordinates of a point of division are (h, 0) where h is the x coordinate.

Now, we know that when two points \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[\left( {{x}_{2}},{{y}_{2}} \right)\] are divided by any point (h, k) in ratio m:n, then we can say, \[h=\dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n}\] and \[k=\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n}\]. So, for points A (1, – 5) and B (– 4, 5), point (h, 0) divides in the ratio m:1, we can write,
\[h=\dfrac{m\left( -4 \right)+1\left( 1 \right)}{m+1}....\left( i \right)\]
And, \[0=\dfrac{m\left( 5 \right)+1\left( -5 \right)}{m+1}....\left( ii \right)\]
Now, from equation (ii), we will find out the value of m. So, we will get,
\[0=\dfrac{5m-5}{m+1}\]
Now, we will cross multiply the equation. So, we will get,
\[0=5m-5\]
\[5m=5\]
\[m=1.....\left( iii \right)\]
So, we get the ratio as 1:1.
Now, from equation (iii), we will put the value of m in equation (i). So, we will get,
\[h=\dfrac{1\left( -4 \right)+1\left( 1 \right)}{1+1}\]
Now, we will simplify it to find the value of h. So, we will get,
\[h=\dfrac{-4+1}{2}\]
\[h=\dfrac{-3}{2}\]
Hence, we can say that the coordinates of the point which divides the line joining A (1, – 5) and B (– 4, 5) in the ratio 1:1 is \[\left( \dfrac{-3}{2},0 \right)\]

Note: In such types of questions, the general mistake which students make is assuming m:1 and at the time of using it in the formula, students tend to use it as 1:m,such mistakes should be avoided and there must be clarity of m,n values.