Proving the Equation of a Slope of a Parabole. Trying to understand an equation of a manual. Explanation of counter-intuitive step.

Question

By trying to prove the equation of the tangent of a parabola i cannot understand a step in the equations.

Tangent of a parabola is calculated with the equation $$yy1 = p(x+x1)$$ and the equation of the parabola is $$y^2 = 2px$$

Assume as in the picture above a tangent that crosses the Parabola in tow points $$ M1(x1,y1)$$ $$M2(x2,y2)$$

The slope will be $$ λ = (y2-y1)/(x2-x1)$$

So we will have a line with the equation: $$ y - y1 = [(y2-y1)/(x2-x1)]*(x-x1)$$ Since M1 and M2 belong to the Parabole we have $$y1^2 = 2px1$$ and $$y2^2 = 2px2$$

Afterwards (and this is the step i cannot understand) he just writes that we have:

$$ y2^2 - y1^2 = 2p(x2-x1) $$

Why? How does he decide to simply subtract the squares? Is there a logical step before that makes this step plausible? He just does it because he knows he can? I just cannot fathom it.

Afterwards until the prove the next steps which i understand are:

$$(y2-y1)(y2+y1) = 2p(x2-x1)$$ $$(y2-y1)/(x2-x1) = (2p)/(y2+y1)$$

So the line of the equation takes the form: $$y-y1 = [(2p)/(y2 + y1)]*(x-x1)$$

or $$(y-y1)(y2+y1) = 2p (x - x1)$$

If M2 it going almost on M1 the coordinates of two are identical so we write:

$$(y-y1)(y1+y1) = 2p (x - x1)$$

$$y1(y-y1) = p (x - x1)$$

$$yy1 = p(x+x1)$$

For the love of me i cannot understand why he subtracts the squares? How does he make the logical step?

Answer 1

We are deriving an equation, but, instead of a single logical sequence of steps, we have the abrupt introduction of a new idea which then "magically" leads to the final result.

This is the kind of thing that puzzled many of my students and left them wondering how they could ever be good at mathematics if this kind of "trick" was required to get a result.

I think this kind of derivation is best taught in a classroom where we can try to imagine what the thinking behind it is.

Suppose we don't know the result. We set out the basic equations as you have done, we get to $y-y1=[(y2-y1)/(x2-x1)]*(x-x1)$

And then we have nowhere to go.

What do we do when we have equations with unknowns? We try to eliminate some by substituting from other equations.

The only other equations we have are for the parabola. We see squares are involved and we realise that the difference of two squares gives us an expression containing $y2-y1$. So let's try that.

And the rest falls out.

We then tidy this all up as a formal proof, losing the thought processes that went into it, and leaving the student pondering how can I ever acquire this kind of insight.

So, to answer your question, the formal proof makes it look like a sudden leap, but the thinking might be no less intuitive than that used in the formal processes for solving simultaneous equations.