Q.4. To find the equation of a circle passing though given points and touching a given circle

Qestion 4:

The centre of the circle passing through (0,0) and (1,0) and touching the circle x^2+y^2=9 is ______?

Answer 4:

Let the required circle be

x^2 + y^2 + 2gx + 2fy + c = 0

As it passes through (0, 0), c = 0

=> its equation is x^2 + y^2 + 2gx + 2fy = 0

As (1, 0) lies on it

1^2 + 0 + 2g = 0 => g = -1/2

=> eqn. of the circle is

x^2 + y^2 - x + 2fy = 0

The eqn. of the common chord of the two circles is

(x^2 + y^2 - x + 2fy) - (x^2 + y^2) = 0

=> x - 2fy = 0

If the circles touch each other, the common chord must be tangent to both the circles

=> perpendicular distance from (0, 0) to the common chord = radius 3

=> l 0 + 0 - 9 l / √(1 + 4f^2) = 3

=> 1 + 4f^2 = 9

=> f^2 = 2

=> f = ± √2

=> centres of the required circles are (1/2, ± √2).

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