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Problem 84 Hard Difficulty

Draw a diagram showing two perpendicular lines that intersect on the y-axis and are both tangent to the parabola $ y = x^2. $ Where do these lines intersect?

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Hey, it's clear. So when you read here, so we have why is equal to X square? This is even a function symmetric around the white access, and we know that are perpendicular lines will have slopes of one and negative one, and they'll go through the y axis. So for the tangent slope of one for why is equal to X square, you have our derivative, which is equal to two X. You cook in one, and we get X is equal to 1/2. We find the white corn in it and it becomes 1/4. We're you know that it's even so have symmetry. So the other is gonna be negative 1/2 and one for we're gonna find the equation With the first slope of one. I get why minus why one is equal to m times X minus X one, and we get why minus 1/4 is equal to one times X minus 1/2 and this becomes a lie is equal to X minus 1/4 for negative one. We're going to do the same thing, and we get why is equal to negative X minus 1/4. We see where the lines intersect by making them equal to one another. When we see that it's Pax is equal to zero under why value becomes negative. 1/4 Put zero comma. Negative. 1/4. We're gonna draw next. Why? Looks like this then. Our tangents. What? This is negative. 1/2 and 1/4 This this positive 1/2 and 1/4 room. This is Hero Common negative 1/4.