Calculus
the equation to the tangent line to the curve x^2 + y ^2 = 169 at the point (5, -12) is A. 5y - 12x = -120 B. 5x - 12y = 119 C. 5x - 12y = 169 D. 12x + 5y = 0 E. 12 x + 5y = 169
The correct answer is C. The easiest way to see this is to realize that the original equation is a circle with center at the origin and radius 13. The point (5,-12) is on the circle (4th quadrant) and forms a perfect 5-12-13 right triangle. If we look at a line through the center of the circle to the point (5,-12) it is by definition perpendicular to the tangent line in question. The slope of the radius line is -12/5. Hence, the slope of the tangent line is 5/12 (the slopes of perpendicular lines are negative reciprocals). From this alone, only B or C could be correct. Since the point (5/-12) must be on the tangent line as well, plugging into C will show that it is indeed a point on the line and the answer to the question.