Help with a question about the tangent to a parabola.

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MichaelOB

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Nov 17, 2019
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The question is:

"Suppose [MATH]y = mx + c[/MATH] is a tangent to [MATH]y^2 = 4x[/MATH]. Show that [MATH]c = 1/m[/MATH], and that the coordinates of the point of contact are [MATH](1/m^2 , 2/m)[/MATH]."

NB. This is a Non-calculator question.

Firstly, to me, the first part of the questions sounds like "Show that for any point on the graph of [MATH]y^2 = 4x[/MATH], the tangent to the graph has a y-intercept equal to [MATH]1/m[/MATH] where m is the gradient at said point". Is this correct?

Now to solving the question: I rearranged [MATH]y^2 = 4x[/MATH] to get [MATH]y = 2sqrt(x)[/MATH] and differentiated so that [MATH]dy/dx = 1/sqrt(x) = m[/MATH]. (I'm not sure how to take the negative root into account after the rearrangement)

After this, I wasn't really sure what I was doing and I wrote: [MATH]y = mx + c = (1/sqrt(x))x +c = sqrt(x) + c[/MATH] and then equalled this to [MATH]2sqrt(x)[/MATH] so that [MATH]2sqrt(x) = sqrt(x) + c[/MATH]. This simplifies so: [MATH]c = 2sqrt(x)-sqrt(x) = sqrt(x)[/MATH] which indeed does agree with the question as [MATH]sqrt(x) = 1/m[/MATH] but I don't think I have actually proved anything.

Please can I have help with finishing the two parts of the question. Maybe I lack some fundamental piece of knowledge which I need assistance in spotting. Thanks.

I also apologize if the formatting is unhelpful, this is my first question
 
The first half of your work is correct, until you tried making the line.

The trouble is that you have to distinguish the particular x at which you are forming the tangent from the general x that remains in the equation of the line. You might call the former [MATH]x_1[/MATH], or just something like [MATH]a[/MATH]. Then [MATH]m = \frac{1}{\sqrt{a}}[/MATH], and you can find the y-intercept given that slope.

The way you formatted equations is fine.
 
Ok, so I understand now the importance of distinguishing between the two variables but once I have [MATH]y = (1/sqrt(a))x + c[/MATH] I cannot substitute random numbers in nor can I make it equal to [MATH]2sqrt(x)[/MATH] as I did in the first post. I'm sorry, I seem to be completely stuck.

PS. How do I write division and square-root signs properly? Using the 'MATH' in square brackets doesn't seem to do the trick.
 
For help with LaTeX math formatting, see https://www.freemathhelp.com/forum/threads/latex-math-formatting-is-available.109843/ . Square roots are written as "\sqrt", and division in the form of a fraction looks like "\frac{a}{b}".

To find c, you have to use the fact that your line not only has slope 1/sqrt(a), but passes through the point (a, 2 sqrt(a)). So you can plug those coordinates into the equation and solve for c. Note that this is actually what you already did, which is why it worked! You just never actually wrote the equation of the line itself.
 
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