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Problem 48 Easy Difficulty

Find an equation for the surface consisting of all points $ P $ for which the distance from $ P $ to the x-axis is twice the distance from $ P $ to the $ yz $-plane. Identify the surface.

Answer

$y^{2}+z^{2}=4 x^{2},$ circular cone

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Video Transcript

Okay. So what do you want to do is we want to find an equation for the surface consisting of all points. Speak. These are the do not know what it is that services extremity says that the distance from P. To take success. So piece of fertilizing the plane we draw. This is my key access. My ex this is my ex, this is my wife. So be somewhere is here. I do not know where it is. So what it says is that let's say this is my feet. The distance from P. To the X. Axis. So the distance from P. E. To the X. Axis. This is a distance because this is X. Y. Z. I do not know what this is. Point is is twice the distance from P. To the why is he planning? So if I look at the distance from P. To the Y. Z. Plane so the distance from Y. T. Plan. So be distance Yeah P. To the wife G. Player. Thank you. So they send from P. To the I. G. Plane is basically the length of. So this is my ex right. This is the place the plane is really X ecology. I'm looking at the distance from X. Y. Z two X equals zero. And then I'm saying that the distance from B to the X axis, distance from the white japan is really the length of X. And is this the same as the twice the distance? So the distance repeated the X axis. So distance from B to the X axis. So if this is my success and the X axis, remember Y Z is zero. So this is X 000 So the distance from P. To the X axis is exactly is basically x minus X square. Last Y zero sq. No g minus is. You don't square. And in respect. And in addition from P to the print is exactly the length of X. So what we're suggesting here is that y square was the square. The square is twice the distance of X. They face square both sites, squaring both sides. We get that Y square plus G square. He's exactly poor access schools. And what is this? This is a circular cone. Yeah. And then how do you know this is a critical Let me draw it for you for the sake of convenience. What I'll do is I'll say that this is my next access. This is why this is Yeah. And similarly here, Yeah. So what happens here is that if X is zero then I'm here. There's no problem. Let's say if X is one, then I am then if X is one, then I'm a circle of radius for so for X equal to one, I'm a circle of radius for so, so I'm something like that readies to and then if X equals two, then I'm similarly circle of radius for now. So, so what happens here really is that this guy really goes like this and joining decision, she will have this side and same thing happens, you fix is negative because they're fixing negative and I'm here but they still come. So this is the superstar. This is a circular.