CircleHard
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
Create a free account to view solution
View Solution FreeMore Circle Questions
x - 2y + 4 = 0 is a common tangent to y2 = 4x & = 1. Then the value of b and the other common tangent are given by -...The number of common tangents to the circlrs x2 + y2 = 4 and x2 + y2 - 6x - 8y = 24 is...A circle has the same centre as an ellipse & passes through the foci F1 & F2 of the ellipse, such that two curves inters...The equation k (x2 + y2) − x − y + k = 0 represents a real circle, if-...If the lines a1 x + b1 y + c1 = 0 and a2 x + b2 y + c2 = 0 cut the coordinate axes in concyclic points, then -...