find the equation of a parabola through (-15/4, 2) and (0, -1) whose axis is parallel to the x-axis and latus rectum equal to 4. The answers are (y-3) ^2 = 4(x+4) and (y+2)^2 = -4(x- 1/4) i have no idea what are those steps in getting the said answers.

Question

psymon · Answer

Well, axis parallel to the x-axis must mean this is a y^2 parabola. Parabolas come in the form of either:
$$x ^{2}=4py$$or
$$y ^{2}= 4px$$
the squared term remains by itself and is always positive. As for the latus, that is the width from one part of the parabola to the other traveling through the focus. The latus rectum is equal to 4p. So since it says the latus rectum is 4, then 4p must be 4. So so far we know this:
$$y ^{2}=4x  $$
Now since you only have two answers to choose from, we find the correct answer by seeing which equation satisfies the given points. So one of your points is (0,-1). This means that if you plug in 0 for x and -1 for y, both sides should come out to be equal. Go ahead and check it :) And then you'll see what the correct answer is.

anonymous · Answer

Thank you but my head's still cracking. are you sure that it would result to (y-3) ^2 = 4(x+4) and (y+2)^2 = -4(x- 1/4)? I am so sorry..

psymon · Answer

Well the answer would be the second one. If you plug in the points (0,-1) and (-15/4, 2), both would cause both sidesof the equal sign to be the same. Or are you unsure of the process of how I got that far?

anonymous · Answer

the (0, -1) are the vertex?

psymon · Answer

No, those are just points that your problem gave. They allowed me to check and be sure which one was the correct equation.

anonymous · Answer

how can I get the (h,k) given the latus rectum and points?

psymon · Answer

I'm not sure how you can. You can get an idea, though. The directrix is at -p units from the center and the focus is at +p units from the center. So in this case, the latus rectum, which is 4p, is 4, meaning p = 1. SInce we are dealing with a parabola that opens to the right, we know that this p = 1 has to do with x-coordinates. Without having the answers there, I do not think you could actually find the center. But since we know the answer, the center is at (-2, 1/4). The focus is +p units from that, so that means we are at (-1,1/4) for the focus and the directrix is -p away, so we are at (-3,1/4) for that. I wish I knew of a way to give you the center from the info given.

anonymous · Answer

Thank you so much Psymon :) great help!