A circle in the xy -plane has its center at - 4 , - 6 . Line k is tangent

Question

A circle in the xy -plane has its center at - 4 , - 6 . Line k is tangent to this circle at the point - 7 , - 7 . What is the slope of line k ?

Accepted Answer

Choice A is correct. A line that's tangent to a circle is perpendicular to the radius of the circle at the point of tangency. It's given that the circle has its center at - 4 , - 6 and line k is tangent to the circle at the point - 7 , - 7 . The slope of a radius defined by the points q , r and s , t can be calculated as t - r s - q . The points - 7 , - 7 and - 4 , - 6 define the radius of the circle at the point of tangency. Therefore, the slope of this radius can be calculated as - 6 - - 7 - 4 - - 7 , or 1 3 . If a line and a radius are perpendicular, the slope of the line must be the negative reciprocal of the slope of the radius. The negative reciprocal of 1 3 is - 3 . Thus, the slope of line k is - 3 . Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. This is the slope of the radius of the circle at the point of tangency, not the slope of line k . Choice D is incorrect and may result from conceptual or calculation errors.

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