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Problem 73 Hard Difficulty

The equation $ x^2 - xy + y^2 = 3 $ represents a "rotated ellipse," that is, an ellipse whose axes are not parallel to the coordinates axes. Find the points at which this ellipse crosses the $ a-axis $ and show that the tangent lines at these points are parallel.

Answer

crosses at $( \pm \sqrt{3}, 0)$ and both tangents have slope 2

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Mtro. C.

July 29, 2021

Your company produces certain gadgets at price of $5 each. Market research indicates that if those gadgets are sold at $15 each, there will be an averange of 200 sales a month, but each dollar increase (or decrease) in the price, there would be 8 fewer (o

Video Transcript

in this problem or given any oppression? Uh, Penelope's. We act first as to show the points that this l observe crosses the X axis. So we're looking for X intercept. Ex intercepts mean that why should be zero and it boy zero people exerting we have excrement. 0003 It means that experts should be treats x to be positive. Negative plus minus started three. And if that is the extent exits accepts the ordinance, they will be negative spirit of +30 in positive skirt off three and zero. Now let's find be 10 gently. So to do that, we're gonna take the everything office or puncture with respect to X, we have two x minus cortical for a second term. We have Why minus accent Boyfriend plus two y y z zero Promise We sit out there into what partners equal to y minus two x You wanna buy to warm months? That's an appeal that disturbance the exit triceps to be Have wife run You let that negative heard of three and zero And what prime Hewlett that was destroyed a B and zero you looked at. They had started a three week under wanted to be to skirted a free to a spirit of free Sudden. It's too, when you left that why prime ah, spirit of really have dated tubes for a feed you at a negative spirit of free. So the answer is again to It means that that doesn't want to Sep. Prejudice lines are parallel. Let's see that be Julie. This is B Ellipse that we would go out with this function. This is the first times have led at first except yourself. And this second transient leg, he's argued, accepts alternative being zero native skirted a 30 And since we found it supposed to be equal, we know that those two are fella.