When doing this problem, remember that cotangent 2 Theta equals a -, C / b. So if we have cotangent 2 Theta equaling 1 -, 4 / -4, we can see that that's -3 / -4 or positive 3/4. So we're going to draw ourselves a triangle. We're going to call this two Theta cotangents adjacent over opposite. So we can see this is a 345 triangle. So we can see sine Theta is going to equal sqrt 1 minus cosine 2, Theta over two, and sine cosine Theta is going to equal sqrt 1 plus cosine 2 Theta over 2. These are both going to be positive because the cotangent was positive. So we get sqrt 1 - 3/5 / 2, square root 1 + 3 fifth over 2. If we multiply top and bottom by 5:00, we'd get 5 + 3 / 10 sqrt 8 tenths or 4/5. So if we rationalize radical 5 over radical 5:00, we'd get 2 square roots of 5 / 5 for the cosine, doing the same kind of thing back here for the sine. If we multiply by 5 / 5, we get 5 - 3 / 10, so square root of 2/10 or 1/5 sqrt 5 / sqrt 5. So we're going to get sqrt 5 / 5. So now we have our X ^2 -, 4 XY plus four y ^2 + 5 root 5 Y -10 equaling 0. We can determine what kind of figure we expect by doing our b ^2 -, 4 AC. If we do, let's go back one our b ^2 -4 AC, so b ^2 is -4 ^2 -4 * 1 * 4, so we're going to get 0. Hence we're going to expect a parabola, and we know that our X is going to equal X prime cosine. Cosine was two root 5 / 5 minus Y prime sine in this case root 5 / 5 and our Y is going to be X prime root 5 / 5 + y prime 2, root 5 / 5. So now we're just going to stick everything in and see what happens. So X ^2 X prime. Let's actually put the coefficient first. Two root 5 / 5 X prime minus root 5 / 5 Y prime squared -4 Two root 5 / 5 X prime minus root 5 / 5 Y prime times root 5 / 5 X prime +2 root 5 / 5 Y prime +5 root five times root 5 / 5 X prime +2 root 5 / 5 Y prime. Oops, did it go oh come on, two root 5 / 5 Y prime. We'll just rewrite it. Root 5 / 5 X prime +2 root 5 / 5 Y prime. And that got us to the five root 5 Y and then -10 = 0. So when we foil this out to root 5 / 5, two times two is 4 root, five times root 5 is five, 4 * 5 is 20 / 25 X prime squared. When we foil this out, we get 2 * 5 or 1020 fifths for the outer, but we're going to get the same thing for the inner. So we're going to end up with -2020 fifths X prime Y prime plus 520 fifths Y prime squared. For this next part. When we foil to root 5 / 5 times, root 5 / 5 is going to give us 1020 fifths times the -4 so -4020 fifths X prime squared. When we do the outside terms, we're going to get a +4 * 5 2020 fifths. So 2020 fifths times -4 is going to be -8020 fifths X prime Y prime. When we do the inner, we're going to get -520 fifths and -5 * -4 is going to give us a +2020 fifths X prime Y prime. And when we do the last terms, we're going to get -1020 fifths -10 * -4 positive 4020 fifths Y prime squared down here. Oh, I forgot this four y ^2 in here, didn't I? Well, let's come back here and add the four times root 5 / 5 X prime +2 root 5 / 5 Y prime squared. So when we foil this, root 5 / 5 X prime is going to be 520 fifths. 520 fifths times 4 is 2020 fifths X prime squared. When we do the outer terms, we're going to get 1020 fifths, but the inner terms is going to be the same. So 2020 fifths times 4 is going to give us 8020 fifths X prime Y prime. The last term squared to root 5 * 2 root 5 is going to be 4 * 5. 2020 fifths 20 * 4 is 8020 fifths Y prime squared. Then we have the plus 5 root five times root 5 / 5, so that's just going to give us 5X prime plus when we distribute this out, we're going to get 10 Y prime -10 = 0. So now if we combine things, 2020 fifth negative, 4020 fifths and 2020 fifths are all going to cancel -2020 fifths -8020 fifths +2020 fifths positive 8020 fifths are going to cancel 520 fifths, 4020 fifths, 8020 fifths. So 45100 and 2520 fifths Y prime squared plus 10 Y prime is going to equal -5 X prime plus ten. We want to have A1 coefficient, so we're going to multiply everything through. But oh, 25125 twenty fifths is really just five Y prime squared. So we're going to divide everything through by 5 Y prime squared +2 Y prime is going to equal negative X prime +2. I need to complete the square so half of the two squared is one. If I add 1 to one side, we're going to add 1 to the other. So we're going to get y + 1 quantity squared, Y prime plus one quantity squared equaling. I'm going to factor out a -1, and we get X prime -3. So our center, our vertices, because it's a parabola, should be three -1. Our four P is going to equal -1. So our P is negative 1/4. This is going to be opening left and right, and we know it's going to open left because it's a negative. So we're going to have a parabola that looks like this. So we know that our focus is going to be 3 -, 1/4. So 2 3/4 -1 our directrix is going to be X equal. 3 1/4 Our axis of symmetry is going to be Y equaling -1.