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Geometry and Trigonometry / Circles Difficulty: Hard

Which of the following equations represents a circle in the xy-plane that intersects the y-axis at exactly one point?

${(x - 8)}^{2} + {(y - 8)}^{2} = 16$

${(x - 8)}^{2} + {(y - 4)}^{2} = 16$

${(x - 4)}^{2} + {(y - 9)}^{2} = 16$

$x^{2} + {(y - 9)}^{2} = 16$

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Explanation

Choice C is correct. The graph of the equation ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ in the xy-plane is a circle with center $(h, k)$ and a radius of length $r$ . The radius of a circle is the distance from the center of the circle to any point on the circle. If a circle in the xy-plane intersects the y-axis at exactly one point, then the perpendicular distance from the center of the circle to this point on the y-axis must be equal to the length of the circle's radius. It follows that the x-coordinate of the circle's center must be equivalent to the length of the circle's radius. In other words, if the graph of ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ is a circle that intersects the y-axis at exactly one point, then $r = |h|$ must be true. The equation in choice C is ${(x - 4)}^{2} + {(y - 9)}^{2} = 16$ , or ${(x - 4)}^{2} + {(y - 9)}^{2} = 4^{2}$ . This equation is in the form ${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ , where $h equals 4$ , $k equals 9$ , and $r equals 4$ , and represents a circle in the xy-plane with center $left parenthesis 4 comma 9 right parenthesis$ and radius of length $4$ . Substituting $4$ for $r$ and $4$ for $h$ in the equation $r = |h|$ yields $4 equals StartAbsoluteValue 4 EndAbsoluteValue$ , or $4 equals 4$ , which is true. Therefore, the equation in choice C represents a circle in the xy-plane that intersects the y-axis at exactly one point.

Choice A is incorrect. This is the equation of a circle that does not intersect the y-axis at any point.

Choice B is incorrect. This is an equation of a circle that intersects the x-axis, not the y-axis, at exactly one point.

Choice D is incorrect. This is the equation of a circle with the center located on the y-axis and thus intersects the y-axis at exactly two points, not exactly one point.