# Write an equation of a line tangent to the given circle at the given point

Circlesparabolashyperbolas and ellipses do not have any inflection point, but more complicated curves do have, like the graph of a cubic functionwhich has exactly one inflection point, or a sinusoid, which has two inflection points per each period of the sine. It's the longest side of the right triangle. Now OA, we don't know the entire side. But the thing that might jump out in your mind is OB is a radius. So the key thing to realize here, since AC is tangent to the circle at point C, that means it's going to be perpendicular to the radius between the center of the circle and point C.

So this right over here is a right angle. Its slope is the derivative ; green marks positive derivative, red marks negative derivative and black marks zero derivative.

## Tangent of a circle example

So x squared plus 9 is equal to So I'm assuming you've given a go at it. Many people contributed. The slope of the secant line passing through p and q is equal to the difference quotient f. So this right over here is a right angle. Now, we clearly know OC. Subtract 9 from both sides, and you get x squared is equal to But the thing that might jump out in your mind is OB is a radius. And so we know that x squared plus 3 squared-- I'm just applying the Pythagorean theorem here-- is going to be equal to the length of the hypotenuse squared, is going to be equal to 5 squared. Conversely, it may happen that the curve lies entirely on one side of a straight line passing through a point on it, and yet this straight line is not a tangent line. This is the case, for example, for a line passing through the vertex of a triangle and not intersecting it otherwise—where the tangent line does not exist for the reasons explained above. And so now we are able to figure out that the hypotenuse of this triangle has length 5. It's the longest side of the right triangle.

Many people contributed. The existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as "differentiability.

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