The coordinates of the point P(x, y) which divides the line segment joining the points A (x1, y1) and B (x2, y2) internally in the ratio l : m
Problem 1 :
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.
Solution:
Given, A(1, -5) and B(-4, 5)
Let the x axis divide the line segment at point (x, 0) in the ratio k : 1.
Using section formula,
So, the required ratio is 1 : 1.
Here, the point of division is P(x, 0).
On comparing x coordinate,
Hence, the required ratio is 1 : 1 and the coordinates of the point of division is (-3/2, 0).
Problem 2 :
In what ratio is the line segment joining A(6, 3) and B(-2, -5) is divided by the x-axis. Also find the coordinates of the point of intersection of AB and the x-axis.
Solution:
Using the section formula, if a point (x, y) divides the line joining the points (x1, y1) and (x2, y2) in the ratio of l : m, then
Let point P on the x axis divides the line segment joining the points A and B the ratio l : m
Consider P lies on x axis having coordinates (x, 0).
Then,
3(x + 2)=5(6 - x)
3x + 6 = 30 - 5x
8x = 24
x = 3
Hence, the coordinates are (3, 0).
Problem 3 :
In what ratio is the line segment joining A(2, -3) and B(5, 6) is divided by the x-axis. Also find the coordinates of the point of intersection of AB and the x-axis.
Solution:
Using the section formula, if a point (x, y) divides the line joining the points (x1, y1) and (x2, y2) in the ratio of l : m, then
Let point P on the x axis divides the line segment joining the points A and B the ratio l : m
Consider P lies on x axis having coordinates (x, 0).
l : m = 1 : 2
Then,
(x - 5)(1) = (2 - x)(2)
x - 5 = 4 - 2x
3x = 9
x = 3
Hence, the coordinates are (3, 0).
Problem 4 :
Find the point P on the x-axis which is equidistant from the points A(5, 4) and B(-2, 3). Also find the area of ΔPAB.
Solution:
Let the point of x axis be P(x, 0) = (x1, y1)
Given, A(5, 4) and B(-2, 3) are equidistant from P.
PA = PB
PA2 = PB2 --> (1)
Distance between two points is
From equation (1)
x2 - 10x + 41 = x2 + 4x + 13
-10x - 4x = 13 - 41
-14x = -28
x = 2
Therefore, coordinates of P is (2, 0).
Now, Area of ΔPAB
Hence, point P is (2,0) and area of ΔPAB is 12.5 square unit.
May 21, 24 08:51 PM
May 21, 24 08:51 AM
May 20, 24 10:45 PM