Equation of the Tangent Line

Proposition (Equation of a Tangent Line) If exists then the equation of the tangent line to the curve at the point is

Example (Equation of a Tangent Line) Find the equations of the tangent lines to the curve that are parallel to the line

    Solution. The line has slope and we use this with the derivative  of to find the Since we have Solving for we get and Therefore, the points of tangency are at and   The tangent lines are found by using where with and We find and respectively. Therefore, the equations of the tangent lines are and Here's is a graph of and the tangent lines:




Example (Equation of a Tangent Line) How many tangent lines to the curve pass through the point ? At which points do these tangent lines touch the curve?

    Solution. All tangent lines through have the form where Since we our looking for the intersection (point of tangency) we eliminate as follows:



Solving for we obtain, Thus there are two tangent lines and they are tangent at the point Here's the graph of the two tangent lines through along with





Example (Equation of a Tangent Line) Find the equations of both tangent lines through the point that are tangent to the parabola

    Solution. All tangent lines through have the form where Since we our looking for the intersection (point of tangency) we eliminate as follows:



Solving for we obtain, and Thus there are two tangent lines and they are tangent at the points and The tangent lines are and Here's the graph of the two tangent lines through along with

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Cite this as:
Equation Of The Tangent Line
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/equation-of-the-tangent-line.html