JEE Advanced | 2014CircleMedium
Question
A circle S passes through the point (0, 1) and is orthogonal to the circles (x - 1)2 + y2 = 16 and x2 + y2 = 1. Then
Options
A.Radius of S is 8
B.Radius of S is 7
C.Centre of S is (-7, 1)
D.Centre of S is (-8, 1)
Solution
Let the circle be x2 + y2 + 2gx + 2fy + C = 0.
It is orthogonal with (x - 1)2 + y2 = 16
∴ 2(-g + 0) = - 15 + C ⇒ - 2g = - 15 + C
It is also orthogonal with x2 + y2 = 1
∴ 0 = - 1 + C ⇒ C = 1
∴ g = 7
This circle passes through (0, 1)
∴ 1 + 2f + 1 = 0 ⇒ f = - 1
∴ The circle is x2 + y2 + 14x - 2y + 1 = 0
(x + 7)2 + (y - 1)2 = 49
∴ centre : (-7, 1) and radius : 7
It is orthogonal with (x - 1)2 + y2 = 16
∴ 2(-g + 0) = - 15 + C ⇒ - 2g = - 15 + C
It is also orthogonal with x2 + y2 = 1
∴ 0 = - 1 + C ⇒ C = 1
∴ g = 7
This circle passes through (0, 1)
∴ 1 + 2f + 1 = 0 ⇒ f = - 1
∴ The circle is x2 + y2 + 14x - 2y + 1 = 0
(x + 7)2 + (y - 1)2 = 49
∴ centre : (-7, 1) and radius : 7
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