Find the ratio in which the y-axis divides the line segment joining the points (5, -6) and (-1, -4). Also, find the coordinates of the point of division.
Answers
Given : line segment joining the points (5, -6) and (-1, -4).
To prove : ratio in which the y-axis divides the line segment
Solution :
Let the line segment A(5, -6) and B(-1, -4) is divided at point P(0, y) by y-axis in ratio m : n
By using section formula :
P(x,y) = [ (m1x2 + m2x1)/m1 + m2 , (m1y2 + m2y1)/m1 + m2]
Here (x1, y1) = (5, - 6) and (x2, y2) = (-1, - 4)
0 = [(m (-1) + n × 5)/(m + n)]
0 × (m + n) = - m + 5n
0 = - m + 5n
m = 5n
m/n = 5/1
Hence the ratio is 5 : 1.
Now ,
y = (5 × - 4 + 1 × (-6)/ (5 + 1]
y = (- 20 - 6)/6
y = - 26/6
y = - 13/3
Hence the coordinates of the point of division is (0, - 13/3).
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By using section formula :
p(x,y) = [ (m1x2 + m2x1)/m1 + m2 , (m1y2 + m2y1)/m1 + m2]
Here:
(x1, y1) = (5, - 6)
(x2, y2) = (-1, - 4)
0 = [(m (-1) + n × 5)/(m + n)]
0 × (m + n) = - m + 5n
0 = - m + 5n
m = 5n
m/n = 5/1
Hence, the ratio is 5 : 1.
Now,
y = (5 × - 4 + 1 × (-6)/ (5 + 1]
y = (- 20 - 6)/6
y = - 26/6
y = - 13/3