Tangent line equations

[khavar]

I need to be pointed in the right direction in order to start understanding tangent line equations. Let's start with finding the tangent line of a parabola. y₀y = 2a(x₀ + x)

There is no fixed radius like a circle, however, each point on the parabola is equidistant from both the directrix and the focus. My thoughts keep returning to this fact but I am not sure which is the next step.


Regarding the equation for a tangent line to a circle:
x₀x + y₀y = r²
I can find the slope of the radius, which is perpendicular to the tangent line. From there I can find the equation of that tangent line, however, it does not clarify how the author of my book derived the above formula, which I would like to understand.

One pre-calculus book with which I am working does not cover tangent line equations, while the other book with which I am working does cover them, however, does not describe how to find these formulas.

Thank you in advance for the help.

 

[HallsofIvy]

What information are given to find the tangent line? If the problem is to find the tangent line to to, say, \(\displaystyle y= ax^2+ bx+ c\), at \(\displaystyle x= x_0\), without using Calculus, then you use the fact that, in order that y= mx+ n be tangent to \(\displaystyle y= ax^2+ bx+ c\) at \(\displaystyle x_0\) not only must they give the same y value, so that \(\displaystyle ax^2+ bx+ c= mx+ n\), \(\displaystyle x_0\) must, in fact, be a double root. We can rewrite that equation as \(\displaystyle ax^2+ (b- m)x+ (c- n)= 0\). That will have \(\displaystyle x_0\) as a double root if and only if the discriminant, \(\displaystyle (b- m)^2- 4a(c-n)\) is equal to 0. That gives the condition that y= mx+ n is tangent to \(\displaystyle y= ax^2+ bx+ c\) at \(\displaystyle x= x_0\). (This is due to Fermat, prior to the development of the Calculus.)

 

[khavar]

What information are given to find the tangent line? If the problem is to find the tangent line to to, say, \(\displaystyle y= ax^2+ bx+ c\), at \(\displaystyle x= x_0\), without using Calculus, then you use the fact that, in order that y= mx+ n be tangent to \(\displaystyle y= ax^2+ bx+ c\) at \(\displaystyle x_0\) not only must they give the same y value, so that \(\displaystyle ax^2+ bx+ c= mx+ n\), \(\displaystyle x_0\) must, in fact, be a double root. We can rewrite that equation as \(\displaystyle ax^2+ (b- m)x+ (c- n)= 0\). That will have \(\displaystyle x_0\) as a double root if and only if the discriminant, \(\displaystyle (b- m)^2- 4a(c-n)\) is equal to 0. That gives the condition that y= mx+ n is tangent to \(\displaystyle y= ax^2+ bx+ c\) at \(\displaystyle x= x_0\). (This is due to Fermat, prior to the development of the Calculus.)

Here is the problem from my book:
The equation of the parabola is y² = 4ax. Prove that the equation of the line tangent to the point p(x0,y0) on the parabola is y₀y = 2a(x₀ + x). (Hint: D = b² - 4ac = 0)

 

[khavar]

Aha! At first I wasn't understanding that at the tangent point, the equations of the line and the of the parabola are equal. I plugged in integers to understand that they share the same y-value. Not yet completely there, a good start. I need to understand why the discriminant would have anything to do with the tangent point.