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If non-zero numbers a, b, c are in H.P., then the straight line x/a + y/b +1/c = 0 always passes through a fixed point. That point is ?

If non-zero numbers a, b, c are in H.P., then the straight line
x/a + y/b +1/c = 0 always
passes through a fixed point. That point is ?

Grade:12

2 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
7 years ago
Hello student,
recall that a , b ,c in harmonic progression then
b = a / ( 1 +d ) , c = a / (1 +2*d )
from b =a /(1+d ) solve for d = (a-b) /b
plug in then c = (ab) / (2a-b)
the equation of the line becomes
x/ a +y /b + (2a-b) / ab
or (x-1)/a + (y+2) /b =0 (1)
If each term is zero then equation (1) always holds.
So:
x- 1 =0
y +2 =0
Or x =1 , y= -2
Or the line always passing the fixed point ( x= 1 ,y =-2 ) .
Thanks and Regards
Shaik Aasif
askIITians faculty
Preeti
15 Points
2 years ago
Since abc are in HP so 
b = 2ac/ a+c
ab + bc = 2ac
Multiply by 1/abc both side
ab/abc + bc/abc - 2ac/abc =0
We get
1/a -2/b +1/c = o
So points are x= 1,y= -2
(1,-2)
 

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