How to Calculate Tangent Lines to Radius

By Robert Schrader
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In mathematical terms, a "tangent" line is a straight line which touches a curve at one and only one point. Tangents to circles are unique in that they form right angles with their corresponding circle's radius. Determining the equation of a line tangent to a function is usually a complicated process which involves calculus. In the case of circles, however, mathematicians have simplified the calculation by pre-deriving an equation to calculate the slope of a line tangent to the circle at a specific point. In others words, you need only find its equation.

Determine the slope of your tangent line using the (x,y) coordinates of the point where the line touches the circle. For circles, whose equations you write in the standard form of "x^2 + y^2 = r^2", where "r" is the circle's radius, you can represent the slope of a tangent line at any point with the equation "m = -(x/y)". If your circle's equation is "x^2 + y^2 = 100", and you want to find the slope of the tangent at "x = 6", plug "x" into the equation and solve for "y", then plug both coordinates into the equation to get your slope: "(6)^2 + y^2 = 100", or "36 + y^2 = 100". Therefore, "y^2 = 100 - 36 = 64", or "y = 8". The slope of your tangent at "x = 6" is therefore "-(6/8) = -3/4".

Plug your slope and coordinates into the equation "m = (y - y0)/(x - x0)", where "m" is your slope, and "y0" and "x0" are the coordinates you used to arrive at it. For the example circle, this would be "-3/4 = (y - 8)/(x - 6)".

Simplify your equation so it exists in the form "y = mx + b". For the example equation, do this as follows: "-3/4(x - 6) = [(y - 8)/(x - 6)](x - 6)"; or "-3/4x + 18/4 = y - 8, and -3/4x + 18/4 + 8 = y - 8 + 8", so "y = -3/4x + 18/4 + 8 = -3/4x + 18/4 + [8*(4/4)] = -3/4x + 18/4 + 32/4 = -3/4x + 50/4 = -3/4x + 12.5". The equation of the line tangent to the circle "x^2 + y^2 = 100" at the point (6,8) is therefore "y = -3/4x + 12.5".

About the Author

Robert Schrader is a writer, photographer, world traveler and creator of the award-winning blog Leave Your Daily Hell. When he's not out globetrotting, you can find him in beautiful Austin, TX, where he lives with his partner.