Q.18. Two circles.

Qestion 18.

If the circle S(1) :x^2+y^2=16 intersect another circle S(2) of radius 5 in such a manner that the common chord is of maximum length and has a slope equal to 3/4 ,then the coordinates of the centre of S(2) are_______

Answer 18.

If two circles of different radii have common chord of maximum length, its length is the diameter of the smaller circle. Here, it is x^2 + y^2 = 16 and the centre of this circle passes through the common chord.

=> eqn. of the common chord having slope 3/4 and passing through (0, 0) is

y = (3/4)x

The centres of circles S(2) = 0 lies on a line perpendicular to y = (3/4)x and passing through (0, 0)

=> they lie on the line y = - (4/3)x or 4x + 3y = 0

The centres are at a distance = √(5^2 - - 4^2) = 3 from (0, 0) on the above line having slope = - 4/3

tanθ = - 4/3 => sinθ = 4/5 and cosθ = - 3/5

Points at a distance of 3 units from (0, 0) on a line having slope - 4/5

= [0 + 3cosθ, 0 + 3sinθ] and [0 - 3cosθ, 0 - 3sinθ]

= [3*(- 3/5), 3*(4/5)] and [- 3*(- 3/5), - 3*(4/5)]

= (- 9/5, 12/5) and (9/5, - 12/5).

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