1. ## Precalculus Parabola Question

Prove that for a parabola of the form x2 = 4py, for ANY focal chord, lines passing through either endpoint of the focal chord and the vertex will pass through the directrix at the foot of the perpendicular line that can be dropped from the opposite focal chord endpoint to the directrix.

Your proof should use variables to show that it holds for any choice of focal chord.

I know that for any focal chord of any length, one endpoint's coordinates are (x,y) and (-4p^2/x,p^2,y). But how can I use that to solve this question (assuming I even need it)?

Basically, I need to prove that the red and blue line both hit the directrix at the same point, the blue line passes through one endpoint of the (green) focal chord and the vertex, the red line is perpendicular to the green focal chord.

2. ## Re: Precalculus Parabola Question

Originally Posted by mathstudent1235
Prove that for a parabola of the form x2 = 4py, for ANY focal chord, lines passing through either endpoint of the focal chord and the vertex will pass through the directrix at the foot of the perpendicular line that can be dropped from the opposite focal chord endpoint to the directrix.

Your proof should use variables to show that it holds for any choice of focal chord.

I know that for any focal chord of any length, one endpoint's coordinates are (x,y) and (-4p^2/x,p^2,y). But how can I use that to solve this question (assuming I even need it)?

Basically, I need to prove that the red and blue line both hit the directrix at the same point, the blue line passes through one endpoint of the (green) focal chord and the vertex, the red line is perpendicular to the green focal chord.
Here is how I would start it. I'd name things.

$\left(a, \dfrac{a^2}{4p}\right)\ is\ the\ left\ endpoint\ of\ the\ focal\ chord.$

$\left(b, \dfrac{b^2}{4p}\right)\ is\ the\ right\ endpoint\ of\ the\ focal\ chord.$

Now I would start writing down information implied by the problem. For example, what are the co-ordinates of the focus? So what is the equation of the focal chord? What is the equation of the line joining the right endpoint to the vertex? Where does that line intersect the directrix? What is the equation of the line from that intersection to the left endpoint? What is the relationship between the slopes of two perpendicular lines. At this point I do not know which information may be relevant and which irrelevant. I just want to see it all and not forget any of it. Once I have the pieces of the puzzle in front of me, then I can begin to think.

3. ## Re: Precalculus Parabola Question

Originally Posted by JeffM
Here is how I would start it. I'd name things.

$\left(a, \dfrac{a^2}{4p}\right)\ is\ the\ left\ endpoint\ of\ the\ focal\ chord.$

$\left(b, \dfrac{b^2}{4p}\right)\ is\ the\ right\ endpoint\ of\ the\ focal\ chord.$

Now I would start writing down information implied by the problem. For example, what are the co-ordinates of the focus? So what is the equation of the focal chord? What is the equation of the line joining the right endpoint to the vertex? Where does that line intersect the directrix? What is the equation of the line from that intersection to the left endpoint? What is the relationship between the slopes of two perpendicular lines. At this point I do not know which information may be relevant and which irrelevant. I just want to see it all and not forget any of it. Once I have the pieces of the puzzle in front of me, then I can begin to think.
I have everything but the red part. I tried using the perpendicular thing which eventually would lead to a system of equations that resulted in y=-p as the y-answer (as that is the directrix equation). I also tried the distance formula, but that failed. How should I proceed? I have no idea what is relevant and what isn't.

4. ## Re: Precalculus Parabola Question

if you have a vertical parabola centered at x=0, w/directrix line y=-p, then your focus is at (0,p)

5. ## Re: Precalculus Parabola Question

Originally Posted by mathstudent1235
I have everything but the red part. I tried using the perpendicular thing which eventually would lead to a system of equations that resulted in y=-p as the y-answer (as that is the directrix equation). I also tried the distance formula, but that failed. How should I proceed? I have no idea what is relevant and what isn't.
If you already have everything, then what is the ratio of the slope of the focal chord and the line joining one endpoint and the intersection of the directrix with the line running from the other endpoint through the origin? What does that ratio mean?