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Numerade Educator

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Problem 53 Hard Difficulty

How many tangent lines to the curve $ y = x/(x + 1) $ pass through the point (1, 2)? At which points do these tangent lines touch the curve?

Answer

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

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Video Transcript

hands Claire's. So when you right here. So the equation of the lying that passes through one common too with the slope M is gonna be warren minus two over X minus one is equal to him. This gives us why is equal to M X minus M plus two. So attention online and meets the curve at only one point. So it has 10 We're going to set this equation equal to X over X plus one only access equal to and x square minus X plus two X waas and thanks minus M plus two. When we simplify this, get them X square. Plus, that's minus, um, plus two is equal to zero. And for the equation to have only 10 the discriminative has to be zero. So the discriminative off D must have equal to zero. You get one minus for, um, times to minus M. There's equal to zero. We used a quadratic formula. Get em is equal to two plus or minus square root of three over two. Now, if we look at the equation, um, X square plus X minus M plus two is equal to zero. We're gonna use the quadratic formula when we get X is equal to negative one plus or minus the square of the discriminatory over to em. The discriminative zero. And we got two plus or minus square root of three over two, so it gives us a one over squirt. Negative, too, was our minus square root of three. Ridge is gonna substitute to find that why hornet? This gives us a new one over negative one plus or minus square of three. So there's two tensions that passed through and the tensions touched at one over. Negative to Mina Square root of three on my one over negative one fine a square root of three and one over negative two plus scrambled of three coma, one over negative one plus heard of three.