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Prove that the points A(a,o).B(0,b),C(1,1) are collinear if 1/a+1/b=1.

Prove that the points A(a,o).B(0,b),C(1,1) are collinear if 1/a+1/b=1.

Grade:11

2 Answers

Vikas TU
14149 Points
6 years ago
Let A(a,0) , b(0,b) and C(1,1) be the collinear points
Let B divide the line segment joining AC in ratio of k:1
Then,by section formula,the coordiantes of B would be given by (k+a / k+l , k/k+1)
However,we already know that the coordinates of B is (0,b)
 
So equating, 0 = k+a/k+1 i.e k=-a
Also, b = k/k+1
Substituting k = -a , we get
B= -a / -a+1
1/b = a-1/ a
1/b = 1 - 1/a
1/a + 1/b = 1
Hence proved.
Gagan
11 Points
6 years ago
A(x,0)    B(0,y)    C(1,1)
Ratio-k:1
C(1,1)=[k(0)+x]÷(k+1),[k(y)+1(0)]÷(k+1)
 
x      
1=[k(0)+x]÷(k+1)
k+1=x
 
y
1=[k(y)+1(0)]÷(k+1)
k+1=k(y)
y=[k+1]÷k
 
1/x+1/y=1/(k+1)+k/(k+1)
=(1+k)/(k+1)
=1
 
hence proved

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