The point ( - 4,6 divides the line segment joining the points a 6, 10) and b(3, -8 the ratio is)

Find the coordinates of the points which divide the line segment joining A(– 2, 2) and B(2, 8) into four equal parts.

Let P, Q and R be the three points which divide the line-segment joining the points A(-2, 2) and B(2, 8) in four equal parts.

Case I. For point P, we have

Hence, m1 = 1, m2 = 3
x1 = -2, y2 = 2
x2 = 2, y2 = 8Then, coordinates of P are given by

Case II. For point Q, we have

m1 = 2, m2 = 2
x1 = -2, y1 = 2
and    x2 = 2, y2 = 8Then, coordinates of Q are given by

Case III. For point R, we have

The co-ordinates of a point which divided two points `(x_1,y_1)` and `(x_2, x_2)` internally in the ratio m:n is given by the formula,

`(x,y) = (((mx_2 + nx_1)/(m + n))","((my_2 + ny_1)/(m + n)))`

Here it is said that the point (−4,6) divides the points A(−6,10) and B(3,−8). Substituting these values in the above formula we have,

`(-4, 6) = (((m(3) + n(-6))/(m + n))"," ((m(-8) + n(10))/
(m + n)))`

Equating the individual components we have,

`-4 = (m(3) + n(-6))/(m + n)`

-4m - 4n = 3m - 6n

7m = 2n

`m/n = 2/7`

Therefore the ratio in which the line is divided is 2 : 7

Text Solution

Answer : Therefore, the point ( - 4 , 6) divides the line segment joining the points A ( - 6 , 10) and B ( 3 , - 8) in the ratio 2 : 7 .