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Geometry and Trigonometry / Circles Difficulty: Hard

A circle in the xy-plane has its center at $left parenthesis 16 comma 17 right parenthesis$ and has a radius of $7 k$ . Which equation represents this circle?

${(x - 16)}^{2} + {(y - 17)}^{2} = 49 k$

${(x - 16)}^{2} + {(y - 17)}^{2} = 49 k^{2}$

${(x - 16)}^{2} + {(y - 17)}^{2} = 7 k$

${(x - 16)}^{2} + {(y - 17)}^{2} = 7 k^{2}$

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Explanation

Choice B is correct. The equation of a circle in the xy-plane can be written as ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ , where the center of the circle is $(h, k)$ and the radius of the circle is $r$ . It’s given that this circle has a center at $left parenthesis 16 comma 17 right parenthesis$ and a radius of $7 k$ . Substituting $16$ for $h$ , $17$ for $k$ , and $7 k$ for $r$ in ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ yields ${(x - 16)}^{2} + {(y - 17)}^{2} = {(7 k)}^{2}$ , or ${(x - 16)}^{2} + {(y - 17)}^{2} = 49 k^{2}$ . Therefore, the equation that represents this circle is ${(x - 16)}^{2} + {(y - 17)}^{2} = 49 k^{2}$ .

Choice A is incorrect. This equation represents a circle with radius $7 StartRoot k EndRoot$ , not $7 k$ .

Choice C is incorrect. This equation represents a circle with radius $StartRoot 7 k EndRoot$ , not $7 k$ .

Choice D is incorrect. This equation represents a circle with radius $StartRoot 7 EndRoot k$ , not $7 k$ .