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July 4th, 2015, 12:28 PM   #1
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finding equation tangent to an equation

If I have an equation y =f(x)

and I know the coords (x1,y1) of a point that does not lie on the line y =f(x).

If I draw a straight line from (x1,y1) then at what point on f(x) will the staright line be perpendicular to a tangent of f(x)

and hence whats the equation of this tangent in terms of the original point x1,y1?

Thank you
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July 4th, 2015, 01:03 PM   #2
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Quote:
Originally Posted by DarkLink94 View Post
If I have an equation y =f(x)

and I know the coords (x1,y1) of a point that does not lie on the line y =f(x).

If I draw a straight line from (x1,y1) then at what point on f(x) will the staright line be perpendicular to a tangent of f(x)

and hence whats the equation of this tangent in terms of the original point x1,y1?

Thank you
You state that $\displaystyle (x_1,y_1)$ does not lie on the line $\displaystyle y=f(x)$.

Lines have no tangents. So, as stated, your problem makes no sense. Think about it and re-phrase it.
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July 4th, 2015, 01:39 PM   #3
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the equation f(x) has tangents. You've understood what I said wrong. The point (x1,y1) does not indeed lie on f(x). Im wondering this... if I draw a straight line from (x1,y1), at what points does it intersect a tangent of f(x) that is also perpendicular to the line from (x1,y1).

1) the line intersects f(x)
2) at the point that the tangent to f(x) is perpendicular to the line from (x1,y1)

Last edited by DarkLink94; July 4th, 2015 at 01:47 PM.
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July 4th, 2015, 02:15 PM   #4
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Yes, he understood what you meant to say, but that's not what you said!

First, you said "the line y= f(x)". If f is not a linear function of x, that is a curve not a line. If f is linear, then it is a line but lines do not have "tangents" (unless you consider the line its own "tangent"). If y= f(x) is a general curve then there may be a number of places at which a line through $\displaystyle (x_1, y_1)$ is perpendicular to the curve. For example, if $\displaystyle y= \sqrt{1- x^2}$, and $\displaystyle (x_1, y_1)= (0, 0)$, then every line through (0, 0) is perpendicular to a tangent of y= f(x).

You can try to do this: Let $\displaystyle (x_0, y_0)$ be the point on the graph of y= f(x) at which the line through $\displaystyle (x_1, y_1)$. The slope of that line is $\displaystyle \frac{y_1- y_0}{x_1- x_0}$ while the slope of the line perpendicular to the graph at $\displaystyle (x_0, y_0)$ is $\displaystyle -\frac{1}{f'(x_0)}$. So we must have $\displaystyle \frac{y_1- y_0}{x_1- x_0}= -\frac{1}{f'(x_0}$. That is a single equation with two unknowns, $\displaystyle x_0$ and $\displaystyle y_0$ so there may be many solutions.
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July 4th, 2015, 02:19 PM   #5
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Originally Posted by DarkLink94 View Post
You've understood what I said wrong.
You said "the line $\displaystyle y=f(x)$". So I did understand what you said. You just didn't say what you meant to say ... namely "the curve $\displaystyle y=f(x)$".

I now think I understand what you want.

You are given a curve $\displaystyle y=f(x)$ and a point $\displaystyle (x_1,y_1)$ not on the curve. What you want are the coordinates $\displaystyle (x_2,y_2)$ of a point on the curve so that the line through the two points is perpendicular to the tangent at $\displaystyle (x_2,y_2)$. Solve these two equations simultaneously:

$\displaystyle y_2=f(x_2)$

$\displaystyle f^{ \prime}(x_2) = \dfrac{x_1-x_2}{y_2-y_1}$
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July 4th, 2015, 02:32 PM   #6
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thank you very much- exactly what I wanted.

Yes I did say f(x) was a line...

obviously wrong- apologies.
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