Find the point (s) of intersection between 7 =< t _ 2,0 , t > and the cylinder Y = 20 _ 2....

Find the point on the parabola $x=t, y=t^{2},-\infty<t<\infty$
closest to the point $(2,1 / 2) .$ (Hint: Minimize the square of the
distance as a function of $t .$ )

This question given the question with July's So have the I won t equals you that you won. One does that the times when the Manus father or one and we have the aren't you? I use the ask as a variable now with a month for 15 plus when the x times that you want, my honest you. Now let's try to find the intersection between them. We need to put them Ekho. So listen implies that we need to solve the system in question now to minus 40 must equal to minus four. Plus you ask one plus zero d must equal to, you know one plus asked one plus the most ico Hugo the five months to ask. Now notice that from the second question, we can have the ask me go to the 0 100 see coaches zero. From here we can find one plus t equal to five. Therefore, to equal to the far. So then we use these two of them here we put into the first question It means that we should get that u minus four times four equals U minus four plus two times zero. This one You get equal to the minus 14 doesn't get equal to minus two. So clearly they're not equal here. Therefore, we have that you, like one G does not intersect the activity.

Wear given the equation with utilized the first lines on 13 Ego junior Uh, minus 11 Presti. Times too far. There are two s equal to the 21 plus the at times the minus one. Six on. Then we want to find the intersection between them. So we need to put our won t need go to the, Aren't you ask? Doesn't implies that we have the system make questions. Here we have a minus one plus duty must equal generate u minus s. And then the one plus 40 was ico during the one plus six. Asked now to serve the system in question Yeah, we were from the first question Can kinda asked if we go to the three Manus tooty and then we use this form the asked. We're looking to the second equation here. Therefore, we should get the one plus 40 equal to one plus six times three minus duty began a one plus 40 equal to one plus 18 minus 12 t. So it means that we have the 16th tee and we equal thio the 18. 30 equal to the 18 hour 16 r t. We go to the nine out eight, We're gonna t equals United eight. We're looking to the first equation here. So 109 after eight get you Go to the minus 11 last night after eight times too far. So we get equal to the minus one plus night at the far then one plus night out of a Jew. So it was 75. Get equal Jew there minus from last night could you? Five out of four. This wouldn't get equal June 11 out of two.

Okay, so we're given line one Skeptical, too. Why secretive ill and the vehicle to go on line two. That's a good exit. Good to 10 plus T comma y is equal to to my security is the question. I get three in Coast Region now. What do we need to find? Well, we need to find the intersection point of the X axis with a line of the Parametric equations. Okay, so let's substitute the expressions of X y Z in terms of teeth. So we have that two minutes to thine is the cookie dough. So that's why is it contagious? And she is a British. That's thinking three plus three team if he could develop and we find that too addicted to Tootie, so TV put the one. So from this, we can draw that our line. It's usual in to that 10 plus one that's 11 comma to minus two that they're on three *** demon Blustery that still so get 11 common Zero comes up as our intersection point

Okay, so we're given following equation, and, uh, we wanna find the minimum distance to a specific point. And so the distance to the point if we write it will be X minus four squared. Plus why minus two squared plus the Z squared. All right. And so let's just let f equal this insight, because if you minimize the insight, we'll get the same thing. So we need to find ingredient of af and ingredient of key. Yeah, All right. So the greedy and of f just find their persons to redraw that city, so it looks like a zero. So the partial introspect X in that direction is going to be So you just bring the two tail, so we'll play through two x my a seat. Bring the two. Don't So too. Why minus four? That you don't choosy. All right. And so our partials here for GM, this is G M. We're going to be so right over here. G x g y Jean z. So any jokes do you want negative to Z, right? So this is going to be equal. Thio, Linda times she works. This is going to be equal to win the times to buy. This is going to be equal to negative, Linda. Times twos years. All right, so we have to z on both sides here, so we have the is equal to negative one. So let's put that back in. So we have two X here. Negative two x negative two y. So if we want Thio, bring this over to this site we're going to get four x minus. Eat is equal to zero. And bring this over to this sites or just setting this equal to zero. We're going at four y minus four people. This hero. So here you can solve by bringing it over. Just relays the nexus to because two times four is eat and then why is just one? And so it's sort of not possible. So for Z using this year, So we're gonna have toe go back to G and plug in or points there. Um, so we're going to have to squared plus one squared minus Z squared is equal to zero. So five blindness is you squared equals zero. So Z is equal to closer minus the square root of five. Right? So we have Z. It's equal to Let's remind us the square to five, all right, and so we have our point, and that will be to one plus or minus the square root of five.