In Exercises 17–30, find the standard form of the equation of eac...

Question

In Exercises 17–30, find the standard form of the equation of each parabola satisfying the given conditions.Focus: (- 5, 0); Directrix: x = 5

Accepted Answer

Hello everyone in struggle. We're going to be looking at this practice problem where we want to write the equation of the problem in standard form given that it has a focus at negative 11 and zero and the direct tricks at X is equal to 11. So in order to solve this problem I'm going to do a picture to sort of visualize what's going on. So I'm gonna do the y axis sort of short and the X axis longer and I don't want to be super precise, I just want to sort of visualize it. So I'm just going to do or label two points. I'll do negative 11 on the X axis and positive 11 on the X axis. And I'm going to use these two points to graph What I'm given. So I have a focus at -11 and zero. So I'm gonna go ahead and draw a dot at that point and the direct tricks is at X is equal to 11. So I'm gonna do a dotted line across all the points where X is equal to 11 and I'll label that X equals 11. And I just know that that's my direct tricks. So because I have the focus on the left and the direct tricks on the right, I know that my proble is going to open up towards the left because the Parabola opens up towards the focus and opposite of the direct tricks um I don't know the exact position of it yet, but I know that it opens up to the left so it could look like either one of these or none of them. But I know that it opens up to the left. So because I know that it opens up to the left or has an access of cemetery on the X axis. Because you can see if I have a problem that opens to the left. The access of cemetery falls along the X axis. Um I know that that follows the standard equation why minus H squared is equal to four P times X minus K. And from here you can see that there's three variables that we need to solve H. P and K. And thankfully if I find the vertex the vertex actually gives me both H and K. Because the vertex is at H and K. And in this case the vertex is Halfway between the direct tricks and the focus. And if we look at this picture you can see that the direct tricks is at 11 and the focus is at -11. So naturally the vertex would be at this zero point but we can go ahead and also sulfur it mathematically. So for my vertex it lies between two points. I'm going to add them together and divide by two. So I have negative 11 for my focus and positive 11 for my direct tricks I'll divide that by two because I want to get that midpoint and my Y value is the same as the Y value of the focus. If we follow this line you can see the focus and the vertex are in the same place. So like my focus. My valley was also zero. And if you solve this first part you also get zero divided by two. So my vertex is 00 as we New from the graph. So now that we know that the verse text is at 00 I can say that my Parabola will look something like this. Opening up to the left. And with this vertex we now have a church and K. So I can plug that into my equation. So now I have y minus zero squared is equal to four p Times X zero. I simplify that. I have y squared is equal to four p times x. And now you can see that I only need to find the value of P. And P recall is the distance between the vertex and the focus. So in this case to go from the vertex to my focus I need to move negative 11 spots. So P would be equal to negative 11. Again, we can determine this mathematically if I go and get The x. value of my vertex, zero Plus the x value or the x value of my focus which is negative I get negative 11 because 0 - is negative 11. So I can also plug that into my standard equation. So this becomes why squared is equal to four times negative 11 times X. And when I put that all together, I have y squared is equal to four times negative 11 is negative 44 X. So this becomes my final answer. You can also jump to this answer directly. If you recall that when the focus is in the form A zero, Then the equation for the problem should be why squared is equal to four a. x. And in this case our focus was negative 11 0. So you can see that A is equal to negative 11. If we plug that in, we got y squared is equal to four times negative X. Which is y squared is equal to negative 44 X. So we got the same answer as a long way. So this could be a shortcut if you're able to recognize that the focus followed the same form, but at the end of the day it's the same answer. So whatever works better for you and this aligns with answer choice A. So hopefully this video helped and see you next time

Struggling with College Algebra?

In Exercises 17–30, find the standard form of the equation of each parabola satisfying the given conditions.Focus: (- 5, 0); Directrix: x = 5

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