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

If the circles

x^{2} + y^{2} + 2 a x + cy + a = 0

and

x^{2} + y^{2} - 3 a x + d y - 1 = 0

intersect in two distinct points P and Q then the line

5 x + b y - a = 0

passes through P and Q for

(A)exactly one value of a
(B)no value of a✓ Correct
(C)infinitely many values of a
(D)exactly two values of a

Answer: (B)

Solution

The common chord of the two circles is given by their difference

S_{1} - S_{2} = 0

, which here is

5 a x + (c - d) y + (a + 1) = 0

. For the stated line

5 x + b y - a = 0

to be this common chord, corresponding coefficients must be proportional:

\frac{5 a}{5} = \frac{a + 1}{- a}

, giving

- a^{2} = a + 1

, that is

a^{2} + a + 1 = 0

. Its discriminant is

1 - 4 = - 3 < 0

, so no real value of

a

lets the line pass through both points of intersection.

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

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