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Numerade Educator

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Problem 23 Medium Difficulty

Find the points on the ellipse $ 4x^2 + y^2 = 4 $ that are farthese away from the point $ (1, 0) $.

Answer

$\left(-\frac{1}{3}, \pm \frac{4}{3} \sqrt{2}\right)$

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Video Transcript

we're asked to find the points on the Ellipse four x squared plus y squared equals four that are farthest away from the point 10 Yeah, To do this was this is good. This is this is it from somebody? First, let's find the distance between this point and the point on the lips. So this distance, which is a function of X and Y d f x y this is the square root of X minus one squared plus why minus zero squared. And so, as a function of X d F X is equal to kind of well, on the one hand, the square root of X minus one squared plus y squared, which we know is four minus four x squared. So you got some more? You have stuff. Mhm. That's the worst part of that now, because D is strictly more greater than or equal to zero and D is increasing. It follows the d. Yeah, is minimized. It was the guy recording that you picked up his responses every job in his lap when the function F, which is D squared, is minimized as well. Uh huh. Now, ffx, this is d squared of X, which is X minus one squared plus four minus four x squared. To find the minimum of f, I'm going to take the derivative and playing critical values. So the derivative of prime of X this two times X minus one minus eight x It's fine. Critical values set this equal to zero. Yeah, so I have two X minus eight X is negative. Six x Yes. Minus two equals zero. So X equals negative. One third. Mhm. Now I'm going to find the second derivative F double prime of X. So this is two minus eight or negative six, which is always less than zero. This is the story of Spin and Matt Small Best Pink. What? Just the New York I know. They have all these. Like I get these headlines are like buying shoes Nigeria Quiet. You know what? Our right. Determined to not be held extreme now because I'm sorry. Instead of minimized, I mean maximized here in the 19% 35 journalists 82 when f was maximized. Okay, Now, because our second derivative is less than zero, it follows that our function f has a relative and because it's the only maximum an absolute maximum at X equals negative one third. So therefore, the points on the lips that are farthest within the origin we can find. Why so why? I swear it is four minus four x squared. This is four minus four times 1/9 which is 36 minus four is 32 9th and therefore, why equals plus or minus a squared of 32. This is four times route to over three plus or minus four thirds route to. So the points on the lips their farthest away from the 0.10 are negative one third plus or minus four thirds route to, like, fucking titties Got