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How many common tangents does the circles $x 2 + y 2 + 2 λ x + c = 0, x 2 + y 2 + 2 μ y − c = 0$ have?

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Zero

One

Two

Three

answer is C.

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Detailed Solution

The general equation of the circle in two variables is

x 2 + y 2 + 2 g x + 2 f y + c = 0

. The centre of the circle is

(− g, − f)

, and the radius of the circle is

r = g 2 + f 2 − c

.
Let the given equation of the circles be,

x 2 + y 2 + 2 λ x + c = 0

… (1)

x 2 + y 2 + 2 μ y − c = 0

… (2)
Compare the equation (1) and the general equation, and we get

g = λ f = 0

The centre and radius of the circle (1) is,

C 1 (− g, − f) = C 1 (− λ, 0) r 1 = (− λ) 2 + 0 − c = λ 2 − c

Compare the equation (2) and the general equation, and we get

g = 0 f = μ c = − c

The centre and radius of the circle (2) is,

C 1 (− g, − f) = C 1 (0, − μ) r 1 = 0 + (μ) 2 − (− c) = μ 2 + c

We know that the square of distance between two centres of two circles is the sum of squares of the radii (i.e.) circles that intersect orthogonally.
So, the distance between the circles is given by,

C 1 C 2 = (0 − (− λ)) 2 + (− μ − 0) 2 = λ 2 + μ 2

The sum of the squares of the radii is given by,

r 12 + r 22 = λ 2 − c 2 + μ 2 + c 2 = λ 2 − c + μ 2 + c = λ 2 + μ 2 = (C 1 C 2) 2

Therefore, the circles intersect orthogonally. We know that the interesting circles have two common tangents.
Hence, the correct option is 3.

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