Circle 4 Question 3

3. Two circles with equal radii are intersecting at the points (0,1) and (0,1). The tangent at the point (0,1) to one of the circles passes through the centre of the other circle. Then, the distance between the centres of these circles is

(2019 Main, 11 Jan I)

(a) 2

(b) 22

(c) 1

(d) 2

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Answer:

Correct Answer: 3. (d)

Solution:

  1. Clearly, circles are orthogonal because tangent at one point of intersection is passing through centre of the other.

Let C1(α,0) and C2(α,0) are the centres.

Then, S1(xα)2+y2=α2+1

S1x2+y22αx1=0

[ radius, r=(α0)2+(01)2]

and S2(x+α)2+y2=α2+1

S2x2+y2+2αx1=0 Now, 2(α)(α)+200=(1)+(1)α=±1

[ condition of orthogonality is 2g1g2+2f1f2=c1+c2 ]

C1(1,0) and C2(1,0)C1C2=2



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