AMSCO'S Geometry. New York - Rye High School

588 Geometry of the Circle13-8 TANGENTS AND SECANTS IN THE COORDINATE PLANETangents in the Coordinate PlaneThe circle with center at the origin andradius 5 is shown on the graph. Let lbe a line tangent to the circle atA(3, 4). Therefore, l ' OA since a tangentis perpendicular to the radiusdrawn to the point of tangency. Theslope of l is the negative reciprocal ofthe slope of OA.yP1O1A(3, 4)lxslope ofOA 5 4 2 03 2 05 4 3Therefore, the slope of l 52 3 4. Wecan use the slope of l and the pointA(3, 4) to write the equation of l.y 2 4x 2 3 523 44(y 2 4) 523(x 2 3)4y 2 16 523x 1 93x 1 4y 5 25The point P(1, 7) makes the equation true and is therefore a point on thetangent line 3x 4y 25.Secants in the Coordinate PlaneA secant intersects a circle in two points. We can use an algebraic solution of apair of equations to show that a given line is a secant. The equation of a circlewith radius 10 and center at the origin is x 2 y 2 100. The equation of a linein the plane is x y 2. Is the line a secant of the circle?