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Show that the curve of intersection of the surfaces $ x^2 + 2y^2 - z^2 + 3x = 1 $ and $ 2x^2 + 4y^2 - 2z^2 - 5y = 0 $ lies in a plane.

These are two hyperboloids of one sheet.

Vectors

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So what I saw is that the kind of intersection? Uh Extra square last two Y squared G squared plus three X. Equal to one. And who is this square? Nice. Cool device. Square Find a school minus question zero the line that plane. So we have a couple of options. So it is easy to just separate salt for something which doesn't have a square. So you can take something from here and something from here at all. So here, since X doesn't have a square, I'll just choose that. You can even choose to do it from here as well. And sulfur. Why? But let's do this one. So from here we know that access square lost two Y squared minus G square Last three x equals to one. So that gives us the GTX is one plus G square Minor successful -2 Y spread. So X is one last G square- Success Square -2. I swear upon tree. Uh Okay. So yeah, I think it is easy not do this because it doesn't really this involves that's too much. So I think it is easy to separate and why from here? So from here, investigators are 5, 5 are you know, not just not. Okay, it is easy to separate the G square. So what I'll do is I'll separate Aggie Square from here. So G square from here is access square lost to Wild Square Plus three X -1. And then once I then I'll plug the value of the square here. So full access where plus four y squared minus two, the value of the square is access square plus two Y square Plus TX -1. The value of the square that you found out -5 Y Questions. You know? So to access square Last four Y sq miners to access where -4 Y sq minus 66 plus two -5 while he goes to zero. Cool. So these to access to access our cancels out for widespread four Y squared cancels out. So all we get is there six X plus pipe Y equals to two. Yeah. And what you wanted to say is that this lies in a plane and that's exactly what it is because this is 65 plus five plus two whose lives on the expect actually. And that's it. That's the answer for us. We are good.