Q.5. Locus of center of the circle orthogonal to a given circle and passing through a given point

Question 5:

If a circle passes through the point (1, 2) and intersects the circle x^2+y^2=4 orthogonally, then find the equation of the locus of its centre.

Answer 5:

The condition for the two circles

x^2 + y^2 + 2g1x + 2f1y + c1 = 0 and

x^2 + y^2 + 2g2x + 2f2y + c2 = 0

to intersect orthogonally is

2g1g2 + 2f1f2 = c1 + c2.

 For the circle x^2 + y^2 - 4 = 0, g1 = 0, f1 = 0 and c1 = - 4

If the second circle is x^2 + y^2 + 2gx + 2fy + c = 0,

then g*0 + f*0 = - 4 + c

=> c = 4

The equation of the second circle is

x^2 + y^2 + 2gx + 2fy + 4 = 0

If it passes through (1, 2),

1^2 + 2^2 + 2g*1 + 2f*2 + 4 = 0

=> 2g + 4f + 9 = 0

=> the centre (-g, -f) of the circle satisfies the equation

2x + 4y - 9 = 0 which is the required locus.

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