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 ?

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Accepted Answer

Choice C is correct. It’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’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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