Equations of Tangent Lines Passing Through Point P(5,5) for y = x^2 - 4

In summary: I think it would make more sense to say that the question is poorly worded and ask for help to reword it.
  • #1
courtrigrad
1,236
2
Hello all

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

My solution:
If [tex] f(x) = x^2 - 4 [/tex] then [tex] f'(x) = 2x [/tex]. So

[tex] y - 5 = 10(x-5) [/tex]

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
 
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  • #2
Add a nonzero constant to the equation u've found.Does it still pass through (5,5)??

Daniel.
 
  • #3
no it doesn't. wouldn't there be an infinite amount of tangent lines based on the slope? So would I just write

[tex] y - 5 = 2x(x- 5 ) [/tex] ?

and i just switch signs

I am not sure if you can even represent more than 1 tangent line
 
Last edited:
  • #4
There is only one point namely [tex] P(5,5) [/tex]. So this imply that there is only one tangent line?

Thanks
 
  • #5
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.
 
  • #6
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 [tex] y = x^2 - 5 [/tex] that pass throught the point [tex] P(5,5) [/tex] if there are an infinite amount of lines ?

Thanks
 
  • #7
courtrigrad said:
Hold on

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

Is this correct??
courtigrad said:
[tex] y-5=2x(x-5) [/tex]


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

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

Daniel.
 
  • #8
yes it is correct, because m = 2x. You are given x to substitute in for the equation ( [tex][ P(5,5) [/tex])

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.
 
  • #9
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? :wink: In that case,please relabel one of them with other letter.

Daniel.
 
  • #10
[tex] y - 5 = 2x_1 ( x - 5) [/tex]
 
  • #11
so i guess this is a trick question?
 
  • #12
would 2x-5 be right? and wouldn't 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?
 
  • #13
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
 
  • #14
Yeah 2x-5 is the only one that works
 

FAQ: Equations of Tangent Lines Passing Through Point P(5,5) for y = x^2 - 4

What is the equation of the tangent line passing through the point P(5,5) for y = x^2 - 4?

The equation of the tangent line can be found by taking the derivative of the given function, which is y' = 2x. Plugging in the x-coordinate of the point P (5) into the derivative gives us y' = 2(5) = 10. Therefore, the equation of the tangent line is y = 10x + b. To find the y-intercept (b), we can plug in the coordinates of the point P (5,5) into the equation, giving us 5 = 10(5) + b. Solving for b, we get b = -45. Therefore, the equation of the tangent line is y = 10x - 45.

How do you graph the tangent line passing through the point P(5,5) for y = x^2 - 4?

To graph the tangent line, we can plot the point P(5,5) on the given function's graph. Then, using the slope (m = 10) and the y-intercept (b = -45) found in the previous question, we can plot another point on the tangent line and draw a straight line passing through both points. This will be the graph of the tangent line passing through the point P(5,5).

What is the significance of the point P(5,5) in relation to the tangent line of y = x^2 - 4?

The point P(5,5) is the point of tangency, where the tangent line touches the graph of the function y = x^2 - 4. This means that the slope of the tangent line at this point is equal to the slope of the function at this point, which is 10. The point of tangency is important because it allows us to find the equation of the tangent line and to graph it accurately.

How do you find the equation of the normal line to y = x^2 - 4 at the point P(5,5)?

The normal line is perpendicular to the tangent line, meaning that the slopes are negative reciprocals of each other. Therefore, the slope of the normal line at the point P(5,5) is -1/10. Using the point-slope form, we can find the equation of the normal line as y - 5 = (-1/10)(x - 5), which simplifies to y = (-1/10)x + 5.5. Therefore, the equation of the normal line to y = x^2 - 4 at the point P(5,5) is y = (-1/10)x + 5.5.

How do you determine if a point lies on the tangent line of y = x^2 - 4 at the point P(5,5)?

To determine if a point lies on the tangent line, we can plug the x-coordinate of the point into the equation of the tangent line found in the first question. If the resulting y-coordinate matches the y-coordinate of the given point, then the point lies on the tangent line. For example, plugging in x = 5 into the equation y = 10x - 45 results in y = 10(5) - 45 = 5, which is the same as the y-coordinate of the point P(5,5). Therefore, the point P(5,5) lies on the tangent line of y = x^2 - 4 at the point P(5,5).

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