CircleHard
Question
A circle touches a straight line lx + my + n = 0 and cuts the circle x2 + y2 = 9 orthogonally. The locus of centres of such circles is -
Options
A.(lx + my + n)2 = (l2 + m2) (x2 + y2 - 9)
B.(lx + my - n)2 = (l2 + m2) (x2 + y2 - 9)
C.(lx + my + n)2 = (l2 + m2) (x2 + y2 + 9)
D.none of these
Solution
Let the equation of the circle is -
x2 + y2 + 2gx + 2fy + c = 0 .................(1)
which touches the line lx + my + n = 0
∴
......(2)
and circle (1) is orthogonal to the circle x2 + y2 = 9
∴ 0 × g + 0 × f = c - 9
⇒ c = 9 ......(3)
from (2) & (3)
∴ locus of (-g, - f) is
(lx + my + n)2 = (x2 + y2 - 9) (l2 + m2)
x2 + y2 + 2gx + 2fy + c = 0 .................(1)
which touches the line lx + my + n = 0
∴
......(2)and circle (1) is orthogonal to the circle x2 + y2 = 9
∴ 0 × g + 0 × f = c - 9
⇒ c = 9 ......(3)
from (2) & (3)
∴ locus of (-g, - f) is
(lx + my + n)2 = (x2 + y2 - 9) (l2 + m2)
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