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Choose the correct answer:

In the equation $x 2 + y 2 + 2 g x + 2 f y + c = 0$ represents a circle with X-axis as a diameter and radius a, then which of the following is possible.

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f = 2 a, g = 0, c = 3 a 2

f = 0, g = a, c = 3 a 2

f = 0, g = − 2 a, c = 3 a 2

none of these

answer is C.

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Detailed Solution

The correct answer is option 3.

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

is the equation of the circle.
Transforming the circle in the general form:

x − α 2 + y − β 2 = r 2

x − g 2 + y − f 2 = g 2 + f 2 − c 2

O is the centre of the circle.

O ≡ − g, − f

and the radius is

g 2 + f 2 − c

Put the value

− f = 0 ⇒ f = 0

Here, the radius is ‘a’.
Therefore, the centre is

O ≡ − g, 0

and radius is

g 2 + f 2 − c = a

On solving the radius,

g 2 − c = a 2

Put y=0 in

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

We get,

x 2 + 2 g x + c = 0 ⇒ x = − 2 g ± 4 g 2 − 4 c 2 ⇒ x = − g ± g 2 − c

The possible points are:

f = 0, g = a, c = 3 a 2

and

f = 0, g = − 2 a, c = 3 a 2

We put

c = 3 a 2

g 2 − c = a 2

We get,

g 2 = 4 a 2

⇒ g = ± 2 a

The answer is option 3.

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