Find the equation of the circle concentric with x² + y² – 4x – 6y – 3 = 0 and which touches the y - axis.

Given that we need to find the equation of the circle which is concentric with x² + y² - 4x - 6y - 3 = 0 which touches the y - axis.

We know that the concentric circles will have the same centre.

Let us assume the equation of the concentric circle be x² + y² - 4x - 6y + c = 0. ......- (1)

We know that the value of x is 0 on the y - axis.

Substituting x = 0 in (1), we get

⇒ 0² + y² - 4(0) - 6y + c = 0

⇒ y² - 6y + c = 0 ......- (2)

We need to get only similar roots on solving the quadratic equation (2), since the circle touches y - axis at only one point.

We know that for a quadratic equation ax² + bx + c = 0, to have similar roots the condition need to be satisfied is:

⇒ b² - 4ac = 0

From (2),

⇒ (- 6)² - 4(1)(c) = 0

⇒ 36 - 4c = 0

⇒ 4c = 36

⇒

⇒ c = 9 ..... (3)

Substituting (3) in (1), we get

⇒ x² + y² - 4x - 6y + 9 = 0

∴ The equation of the concentric circle which touches y - axis is x² + y² - 4x - 6y + 9 = 0.