JEE Main · 2005 · 08 May 2005 · Offline · Circle

If a circle passes through the point

(a, b)

and cuts the circle

x^{2} + y^{2} = p^{2}

orthogonally, then the equation of the locus of its centre is

Answer: (D)

Solution

Let the variable circle have centre

(h, k)

and radius

r

. Passing through

(a, b)

gives

r^{2} = (h - a)^{2} + (k - b)^{2}

. Cutting

x^{2} + y^{2} = p^{2}

orthogonally requires

r^{2} + p^{2} = h^{2} + k^{2}

. Substituting the first relation into the second,

h^{2} - 2 ah + a^{2} + k^{2} - 2 bk + b^{2} + p^{2} = h^{2} + k^{2}

, which simplifies to

2 ah + 2 bk = a^{2} + b^{2} + p^{2}

. Writing

(h, k)

(x, y)

, the locus of the centre is

2 a x + 2 b y - (a^{2} + b^{2} + p^{2}) = 0

JEE Main · 2005 · 08 May 2005 · Offline · Single correct option