1998 AHSME Problem 23

Geometry Level 3

The graphs of $x^2 + y^2 = 4 + 12x + 6y$ and $x^2 + y^2 = k + 4x + 12y$ intersect when k satisfies $a \le k \le b$ , and for no other values of k. Find b-a.

5 104 140 144

2 solutions

Adrian Neacșu
Apr 26, 2014

The k can be determined easily if you intersect the first circle with the line that goes through both centers. After you get the points calculate the distance from the center of the second circle to those points. The distances are $2$ and $12$ .

For the second circle the radius is $\sqrt { k}$ , then $a=4$ , $b=144$ .

Ankit Gupta
Apr 18, 2014

radius of both circles is 7 and sqrt(k+40) resp., using the relation b/w the radii and the distance b/w the two centres of circles we get b=104 and a= -36.

1998 AHSME Problem 23

2 solutions

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