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

Question

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

Accepted Answer

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 = 4 , k = 9 , and r = 4 , and represents a circle in the xy -plane with center 4 , 9 and radius of length 4 . Substituting 4 for r and 4 for h in the equation r = h yields 4 = 4 , or 4 = 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 loed on the y -axis and thus intersects the y -axis at exactly two points, not exactly one point.

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