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The value of ​ λ for which the circle x 2 +y 2 -3x+λy-5=0 and 4(x 2 +y 2 )-12x-y-9=0becomes concentric is

The value of​ λ for which the circle x2+y2-3x+λy-5=0 and 4(x2+y2)-12x-y-9=0becomes concentric is

Grade:12th pass

1 Answers

Ram Kushwah
108 Points
3 months ago
GIVEN:
EQUATION OF TWO CIRCLES AS:

 

x²+y²-3x+λy-5=0.......................(1)
 
and 4(x²+y²)-12x-y-9=0
 
Or x²+y²-3x-y/4-9/4=0...................(2)
 
TO FIND:
 
Value of λ If circle (1) and (2) become concentric
 
SOLUTION:
 
The std equation of a circle is:
 
x²+y²+2gx+2fy+c=0...................(3)
centre is (-g,-f)
 
comparing (2) and (3) we get
g=-3/2 ,f=-1/8,c=-9/4
So centre of circle (2) is (-3/2,-1/8)
 
Now comparing (1) and (3) we get
g=-3/2,f=λ/2
 
So centre of circle (1) is (-3/2,λ/2)
 
Circle(1) and (2) become concentric if the centre of both the
circles is same
Thus the centre of circle (1) is (-3/2,-1/8)
On comparision we get
x coordinates are same
and y coordinate
λ/2=-1/8
λ=-2/8=-1/4
 
RESULT:
 
λ = -1/4
 
 

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