We use the section formula to find the point which divides the line segment in a given ratio.
The point P which divides the line segment joining the two points A (x_{1}, y_{1}) and B (x_{2}, y_{2}) internally in the ratio l : m is
If P divides a line segment AB joining the two points
A (x_{1}, y_{1}) and B (x_{2}, y_{2}) externally in the ratio l : m is,
Example 1 :
Find the coordinates of the point which divides the line segment joining (-3, 5) and (4, -9) in the ratio 1:6 internally.
Solution :
Let A (-3, 5) and B (4, -9)
Section formula internally
= (lx_{2}+mx_{1})/(l+m), (ly_{2}+my_{1})/(l+m)
l = 1 and m = 6
= [(1(4)+(6(-3)]/(1+6) , [(1(-9)) + 6(5)]/(1+6)
= (4-18)/7, (-9 + 30)/7
= -14/7, 21/7
= (-2, 3)
Example 2 :
Let A (-6 , -5) and B(-6 , 4) be the two points such that a point P on the line AB satisfies AP = (2/9) AB.
Find the point P.
Solution :
AP = (2/9) AB
9AP = 2(AP+PB)
9AP = 2AP + 2PB
9AP – 2AP = 2PB
7AP = 2PB
AP/AB = 2/7
AP:PB = 2:7
So P divides the line segment in the ratio 2:7
Section formula internally
= (lx_{2}+mx_{1})/(l+m), (ly_{2}+my_{1})/(l+m)
l = 2, m = 7
= [(2(-6)+7(-6)]/(2+7), [(2x(4)+7(-5)]/(2+7)
= (-12-42)/9, (8-35)/9
= -54/9, -21/7
= (-6, -3)
Example 3 :
Find the points of trisection of the line segment joining the points A (2, -2) and B (-7, 4).
Solution :
AP = 1, PQ = 1, QB = 1
= (lx_{2}+mx_{1})/(l+m), (ly_{2}+my_{1})/(l+m)
P divides the line segment in the ratio 1:2
l = 1, m = 2
= [(1(-7)+2(2)]/(1+2), [1(4)+2(-2)]/(1+2)
= (-7+4)/3, (4-4)/3
= -3/3 , 0/3
= P (-1 , 0)
Q divides the line segment in the ratio 2:1
l = 2, m = 1
= [2(-7)+1(2)]/(2+1), [2(4)+1(-2)]/(2+1)
= (-14+2)/3, (8-2)/3
= -12/3 , 6/3
= Q (-4, 2)
Example 4 :
Find the ratio in which x axis divides the line segment joining the points (6, 4) and (1,- 7).
Solution :
Let l : m be the ratio of the line segment joining the points (6, 4) and (1, -7) and let p(x, 0) be the point on the x axis
Section formula internally =
(lx_{2}+mx_{1})/(l+m), (ly_{2}+my_{1})/(l+m)
(x, 0) = [l(1)+m(6)]/(l+m) , [l(-7)+m(4)]/(l+m)
(x , 0) = [l+6m]/(l+m) , [-7l+4m]/(l+m)
Equating y-coordinates
[-7l+4m]/(l+m) = 0
-7l+4m = 0
-7l = -4m
l/m = 4/7
l : m = 4 : 7
So, x-axis divides the line segment in the ratio 4:7.
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