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Grade: 12
        
Given the three circles x2+y2-16x+60=0, 3x2+3y2-36x+81=0,x2+y2-16x-12y+84=0,find (i)the point from which the tangents to them are equal in length and (ii)this length.
3 years ago

Answers : (5)

Nishant Vora
IIT Patna
askIITians Faculty
2467 Points
							let that point be (h.k)

Now we know that Length of tangent is given by root(S1)

So length of tangent to circle 1 is root(h2+k2-16h+60)

Similarily for other two and then equate all the lengths

3 years ago
Hansraj Gyanendra Singh Rajawat
35 Points
							
Can you plz explain it clearly to me to understand?? thanking you Sir..............................................................................................................
3 years ago
mycroft holmes
272 Points
							
The radical axis of two circles has the property that the tangents drawn from a point on it to both the circles will have the same length. Radical axis equation quite naturally is S1=S2 i.e. S1- S2 =0 which is a straight line. 
 
So, for this problem, find the radical axes of any two pairs of circles and their intersection will give you the point you seek( in this problem though, the point is inside the circle and hence length of tangent is not meaningful)
3 years ago
dhruv
12 Points
							
S1: x^2+y^2-16x+60=0
S2:3x^2+3y^2-36x+81=0 =>x^2+y^2-12x+27=0
S3:x^2+y^2-16x-12y+84=0
 
S1-S2=0
x^2+y^2-16x+60-x^2-y^2+12x-27=0
-4x=-33
x=33/4
 
S2-S3=0
x^2+y^2-12x+27-x^2-y^2+16x+12y-84=0
4x+12y-57=0
 
S3-S1=0
x^2+y^2-16x-12y+84-x^2-y^2+16x-60=0
-12y+24=0
-12y=-24
y=24/12
y=2
 
Therefore, (x,y)=(33/4,2)
2 years ago
dhruv
12 Points
							
FIRST SELECT ANY OF THE EQUATION OF THE CIRLE AND THEN PUT THE VALUE OF (x,y)=(33/4)
 
root S1 =>root x^2+y^2-16x+60
          =>root (33/4)^2+(2)^2-16(33/4)+60
          =>root 1089/16+4-132+60
          =>root 1089/16-68
          =>root 1089-1088/16
          =>root 1/16
          =>1/4
2 years ago
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