Answer :

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.


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