Thread: Help proving the formula for a tangent line of a parabola

1. Help proving the formula for a tangent line of a parabola

We started learning about conic sections last week, and our assignment asks us to prove that, for the parabola

y^2=4px

the tangent line at point P(x1, y1) can be written as so:

y1*y=2p(x+x1)

I'm a little confused about how to proceed. I have that the slope will be

dx/dy = y/2p

but am unsure where to go from here.

Thanks!

2. Re: Help proving the formula for a tangent line of a parabola

What you get from dy/dx is the slope of the tangent line to the parabola. Once you find the slope you will need to find the equation of the line of the form y = mx + b where m is the slope dy/dx. Your will find b using the coordinates of P in the equation of the line. You need to calculate the derivative correctly. Refer to your book on how to calculate the derivative.

3. Re: Help proving the formula for a tangent line of a parabola

From the given, you have $\displaystyle 2y\frac{dy}{dx}=4p \Rightarrow \frac{dy}{dx}=\frac{2p}{y}$
Tangent line is given by: $\displaystyle y=mx+b$
Using the parabola equation, point (x1,y1) and the equation of the tangent line, you should be able to show the conclusion.

4. Re: Help proving the formula for a tangent line of a parabola

Thanks very much, guys!

Now I'm trying to find the x-intercept. Using the equation for a tangent line given in the original post,

0 = 2p(x + x1)

2px + 2px1 = 0
2px1 = -2px
x1 = -x

So apparently, the coordinates for the x-intercept are (-x, 0), but the drawing I made makes this hard to believe. I know this is incredibly elementary algebra and whatnot, but I feel like I made a silly mistake somewhere.

5. Re: Help proving the formula for a tangent line of a parabola

Know that $\displaystyle y_1=\frac{2px_1}{y_1}+b$
and $\displaystyle (y_1)^2=4px_1$
Derive the result.

6. Re: Help proving the formula for a tangent line of a parabola

The x-intercept of the tangent line, which as shown, has the equation

y1y = 2p(x+x1)

7. Re: Help proving the formula for a tangent line of a parabola

Originally Posted by RubberDucky
The x-intercept of the tangent line, which as shown, has the equation

y1y = 2p(x+x1)
use the point slope formula: y - y1 = (2p/y)(x - x1)
Multiply both sides by y and rearrange, you will obtain the answer to your problem

8. Re: Help proving the formula for a tangent line of a parabola

Originally Posted by votan
use the point slope formula: y - y1 = (2p/y)(x - x1)
Multiply both sides by y and rearrange, you will obtain the answer to your problem
I get

y^2 = 2px - 2px1
y^2 - 2px = -2px1
x1 = -(y^2 - 2px)/2p

Does that look reasonable? From the wording on my assignment, I'm supposed to be able to easily draw a tangent line to a general point P(x1, y1) using the x-intercept, but that's proving challenging with the derived formulas, and I'm just paranoid that I've done something slightly wrong somewhere.

9. Re: Help proving the formula for a tangent line of a parabola

Originally Posted by RubberDucky
I get

y^2 = 2px - 2px1
y^2 - 2px = -2px1
x1 = -(y^2 - 2px)/2p

Does that look reasonable? From the wording on my assignment, I'm supposed to be able to easily draw a tangent line to a general point P(x1, y1) using the x-intercept, but that's proving challenging with the derived formulas, and I'm just paranoid that I've done something slightly wrong somewhere.

y^2 = 2px - 2px1 <---- what happened to y*y1 term on the left side?

10. Re: Help proving the formula for a tangent line of a parabola

Originally Posted by votan
y^2 = 2px - 2px1 <---- what happened to y*y1 term on the left side?

But at the x-intercept, y1 should be zero, right?

And we've just arrived back at the original equation for the tangent line, so I don't see the relevance. Shouldn't I be solving for x1?

11. Re: Help proving the formula for a tangent line of a parabola

Originally Posted by RubberDucky
But at the x-intercept, y1 should be zero, right?

And we've just arrived back at the original equation for the tangent line, so I don't see the relevance. Shouldn't I be solving for x1?
y1 is the ordinate of P. Why you wanted to set it to 0? The statement of the problem you posted does not say y1 = 0.