Show, using implicit differentiation, that any tangent line at a point p to a circle with center o is perpendicular to the radius op.

A circle centered at with radius (the length of ) has equation

which can be parameterized by

with .

The tangent line to at a point is with and . By the chain rule (and this is where we use implicit differentiation),

At the point , we have

so that the slope of the line tangent to the circle at is

Meanwhile, the slope of the line through the center and the point is

Recall that perpendicular lines have slopes that are negative reciprocals of one another; taking the negative reciprocal of this slope gives

which is exactly the slope of the tangent line.

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step-by-step explanation:

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