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
⇒ (- 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.
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