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Question

If (h, k) is a point from which the tangents to the three circles x2 + y2 4x + 7 = 0, 2x2 + 2y2 3x + 5y + 9 = 0 and x2 + y2 + y = 0 are equal in length. Find the value of h + k ___ .

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Solution

Given circles

s1,x2+y24x+7=0 - - - - - - -(1)

s2,x2+y23x2+5y2+92=0 - - - - - - -(2)

s3,x2+y2+y=0 - - - - - - - (3)

(h,k) ia a point from which the tangents to the three circles are equal in length .So,(h,k) is the radical

centre of the three circles.

So,radical axes of circle (1) and (2) is

S1S2=0

52x52y+52=0

xy+1=0

x+y=1 - - - - - - -(4)

Radical axes of circle (2) and (3) is

S2S3=0

32x+32y+92=0

x+y+3 - - - - - - -(5)

xy=3

Solving equation 4 and 5

We get,

2x=4

x=2

y=1

Coordinates of (h,k) is (2,-1)

Radicalcenter is (2,-1)

h=2,k=1

h+k=21=1


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