A circle in the xy -plane has its center at - 1 , 1 . Line t is tangent to

Question

A circle in the xy -plane has its center at - 1 , 1 . Line t is tangent to this circle at the point 5 , - 4 . Which of the following points also lies on line t ?

Accepted Answer

Choice C is correct. It&rsquo;s given that the circle has its center at - 1 , 1 and that line t is tangent to this circle at the point 5 , - 4 . Therefore, the points - 1 , 1 and 5 , - 4 are the endpoints of the radius of the circle at the point of tangency. The slope of a line or line segment that contains the points a , b and c , d can be calculated as d - b c - a . Substituting - 1 , 1 for a , b and 5 , - 4 for c , d in the expression d - b c - a yields - 4 - 1 5 - - 1 , or - 5 6 . Thus, the slope of this radius is - 5 6 . A line that&rsquo;s tangent to a circle is perpendicular to the radius of the circle at the point of tangency. It follows that line t is perpendicular to the radius at the point 5 , - 4 , so the slope of line t is the negative reciprocal of the slope of this radius. The negative reciprocal of - 5 6 is 6 5 . Therefore, the slope of line t is 6 5 . Since the slope of line t is the same between any two points on line t , a point lies on line t if the slope of the line segment connecting the point and 5 , - 4 is 6 5 . Substituting choice C, 10 , 2 , for a , b and 5 , - 4 for c , d in the expression d - b c - a yields - 4 - 2 5 - 10 , or 6 5 . Therefore, the point 10 , 2 lies on line t . Choice A is incorrect. The slope of the line segment connecting 0 , 6 5 and 5 , - 4 is - 4 - 6 5 5 - 0 , or - 26 25 , not 6 5 . Choice B is incorrect. The slope of the line segment connecting 4 , 7 and 5 , - 4 is - 4 - 7 5 - 4 , or - 11 , not 6 5 . Choice D is incorrect. The slope of the line segment connecting 11 , 1 and 5 , - 4 is - 4 - 1 5 - 11 , or 5 6 , not 6 5 .

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