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Geometry and Trigonometry / Circles Difficulty: Hard

Circle A shown is defined by the equation $x^{2} + {(y - 6)}^{2} = 7$ . Circle B (not shown) has the same radius but is translated $96$ units to the right. If the equation of circle B is ${(x - h)}^{2} + {(y - k)}^{2} = a$ , where $h$ , $k$ , and $a$ are constants, what is the value of $4 a$ ?

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Explanation

The correct answer is $28$ . The equation of a circle in the xy-plane can be written as ${(x - t)}^{2} + {(y - s)}^{2} = r^{2}$ , where the center of the circle is $(t, s)$ and the radius of the circle is $r$ . It’s given that circle A is defined by the equation $x^{2} + {(y - 6)}^{2} = 7$ , which can be written as ${(x - 0)}^{2} + {(y - 6)}^{2} = {(\sqrt{7})}^{2}$ . It follows that $r equals StartRoot 7 EndRoot$ and the radius of circle A is $StartRoot 7 EndRoot$ . It’s also given that circle B has the same radius as circle A. If the equation of circle B is ${(x - h)}^{2} + {(y - k)}^{2} = a$ , then $a = r^{2}$ . Substituting $StartRoot 7 EndRoot$ for $r$ in this equation yields $a equals left parenthesis StartRoot 7 EndRoot right parenthesis squared$ , or $a equals 7$ . It follows that the value of $4 a$ is $4 left parenthesis 7 right parenthesis$ , or $28$ .