CircleMedium
Question
If a circle passes through the point (a, b) and cuts the circle x2 + y2 = k2 orthogonally, then the equation of the locus of its centre is
Options
A.2ax + 2by - (a2 + b2 + k2) = 0
B.2ax + 2by -(a2 - b2 + k2) = 0
C.x2 + y2 - 3ax - 4by + a2 + b2 - k2 = 0
D.x2 + y2 - 2ax - 3by + (a2 - b2 - k2) = 0
Solution
Let x2 + y2 + 2gx + 2 fy + c = 0 cuts x2 + y2 = k2 orthogonally
⇒ 2g1g2 + 2f1f2 = c1 + c2
⇒ -2g.0 - 2 f .0 = c - k2
⇒ c = k2 .....(i)
Also, x2 + y2 + 2gx + 2 fy + k2 = 0 passes through (a, b)
∴ a2 + b2 + 2ga + 2 fb + k2 = 0 .....(ii)
⇒ Required equation of locus of centre of centre is
-2ax - 2by - (a2 + b2 + k2) = 0
or 2ax + 2by - (a2 + b2 + k2) = 0
⇒ 2g1g2 + 2f1f2 = c1 + c2
⇒ -2g.0 - 2 f .0 = c - k2
⇒ c = k2 .....(i)
Also, x2 + y2 + 2gx + 2 fy + k2 = 0 passes through (a, b)
∴ a2 + b2 + 2ga + 2 fb + k2 = 0 .....(ii)
⇒ Required equation of locus of centre of centre is
-2ax - 2by - (a2 + b2 + k2) = 0
or 2ax + 2by - (a2 + b2 + k2) = 0
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