Question 971063: Two circles have their centres on the line y+3=0 and touch the line 3y-2x=0. If the radii of circles are \sqrt{13}
Find the equations of the circle and the coordinates of their centres
Hint use similar triangles.
Answer by anand429(138)   (Show Source):
You can put this solution on YOUR website! Since centres of circles lie on y+3=0, i.e y= -3
Let coordinates of centres be (x,-3)
Since the circles touch the line -2x+3y=0
So, the distance from centre of circles to this line are radii of circles.
So, Using the distance formula i.e
(|-2x+3*(-3)|)/(sqrt((-2^2)+3^2)) = sqrt(13) (|..| = mod sign)
or, |-2x+3*(-3)| = 13
or, -2x-9 = (+-13)
So,  -(i) or  - (ii)
Solving both eqns. separately,
we get,  or 
So, the coordinates of center are (-11,-3) and (2,-3)
Equations of circles are  and 
i.e.  and 
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