prove that if P is a point of the circle C, then there is just one line that is tangent to C at P.

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- May 3rd 2008, 07:41 AMnatesterProof with circles
prove that if P is a point of the circle C, then there is just one line that is tangent to C at P.

- May 3rd 2008, 07:45 AMMoo
Hello,

Do it by a proof by contradiction.

Suppose that there are 2 tangent lines D & D' to C at P.

Let O be the center of the circle.

Then D is perpendicular to OP, and so is D'.

Hence, D & D' are parallel.

Is it possible that two**different**& parallel lines go through a common point (P) ? - May 3rd 2008, 07:49 AMIsomorphism
Depends on how a tangent is defined to you. The equation of the tangent is

$\displaystyle y - y_{P} = \left(\frac{dy}{dx}\right)_{P} (x - x_{P})$

It is clear that $\displaystyle (x_P,y_P)$ and $\displaystyle \left(\frac{dy}{dx}\right)_{P}$ is**fixed**for a point P. Hence the tangent is unique. - May 3rd 2008, 07:58 AMMoo
- May 3rd 2008, 08:00 AMIsomorphism
- May 3rd 2008, 09:22 AMnatester
Both of the responses helped. Thank you