# Tangent Line

Hello all

Find the equations of all lines tangent to $$y = x^2 - 4$$ that pass through the point $$P(5,5)$$

My solution:
If $$f(x) = x^2 - 4$$ then $$f'(x) = 2x$$. So

$$y - 5 = 10(x-5)$$

This is just tthe equation of 1 tangent line. To find all tangent lines would I have to add some constant c to the equation?

Thanks a lot

dextercioby
Homework Helper
Add a nonzero constant to the equation u've found.Does it still pass through (5,5)??

Daniel.

no it doesn't. wouldnt there be an infinite amount of tangent lines based on the slope? So would I just write

$$y - 5 = 2x(x- 5 )$$ ?

and i just switch signs

I am not sure if you can even represent more than 1 tangent line

Last edited:
There is only one point namely $$P(5,5)$$. So this imply that there is only one tangent line?

Thanks

dextercioby
Homework Helper
Though your equation for the tangent LINES is incorrect (the way written,they are not equations for LINES,but for parabolas),i can tell u that the number of tangent lines to a graph in one point is infinite.But from this infinity,only one passes through a fixed point.

Daniel.

Hold on

Why is the above equation incorrect the way it's written? Isn't it correct to use point slope form and find the equation of the tangent line to the parabola? Also, why would the question as: Find the equations of all lines tangent to $$y = x^2 - 5$$ that pass throught the point $$P(5,5)$$ if there are an infinite amount of lines ?

Thanks

dextercioby
Homework Helper
Hold on

Why is the above equation incorrect the way it's written?

Is this correct???????????
$$y-5=2x(x-5)$$

Also, why would the question as: Find the equations of all lines tangent to $$y = x^2 - 5$$ that pass throught the point $$P(5,5)$$ if there are an infinite amount of lines ?

Would you rephrase that??It doesn't make any sense to me... :yuck:

Daniel.

yes it is correct, because m = 2x. You are given x to substitute in for the equation ( $$[ P(5,5)$$)

Hmm, I copied the question exactly the way it was written in the worksheet. Maybe its a trick question, however I am not sure.

Thanks a lot for you help.

dextercioby
Homework Helper
That equation is WRONG.It's for a parabola,not for a tangent line,don't u understand??Or maybe the two "x"-s are not the same??? In that case,please relabel one of them with other letter.

Daniel.

$$y - 5 = 2x_1 ( x - 5)$$

so i guess this is a trick question?

would 2x-5 be right? and wouldnt there only be one if you think of the tangent lines of a parabola as it changes with the parabola it only can passover a point once right?

In my opinion, I think dextercioby is right in saying that this is a poorly worded question. There is no indication of obtaining more than 1 equation for the tangent line.

Thanks to all who helped

Yeah 2x-5 is the only one that works