Circles that touch the axis

Level 2

A circle passes through the point ( 2 , − 4 ) (2,-4) and touches both the x-axis and the y-axis. If the centers of the two circles which satisfy these conditions can be represented as ( a , b ) (a,b) and ( c , d ) (c,d) ,what is the value of a − b + c − d a-b+c-d ?


The answer is 24.

Marc Phua by Marc Phua , 23, Singapore

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1 solution

Ujjwal Rane
Aug 10, 2016

Since it is touching both axes and passing through (2,-4) - a point in the fourth quadrant, the circle must have a center at (R, -R)

So ( 2 − R ) 2 + ( − 4 + R ) 2 = R 2 (2-R)^2 + (-4+R)^2 = R^2 giving R 2 − 12 R + 20 = 0 R^2 - 12R + 20 = 0 solving which we get R = 10 or 2 and the two possible centers would be (10,-10) and (2,-2) giving 10 − ( − 10 ) + 2 − ( − 2 ) = 24 10 - (-10) + 2 - (-2) = \textbf{24}

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