Ano123
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Find the Cartesian equation of the plane  \Pi that the matrix  \mathbf{M}
 \begin{bmatrix} 2 & 1 & 0 \\ 0& 1& 0 \\ 4& 2& 0 \end{bmatrix}
maps all points to under the transformation.
Find the Cartesian equations of the two possible spheres of radii 3 that the plane  \Pi is tangent to at the origin.
[The equations should be in exact form not involving trigonometric functions.]

The last bit is aarrrd, hard to visualise.
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ghostwalker
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(Original post by Ano123)
hard to visualise.
Essentially you have a plane that goes through the origin.

Then, there are two spheres - one either side of the plane - that touch the plane at the origin. You could imagine one sitting on the plane at that point, and the other in the mirror image position - treating the plane as a mirror.
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RDKGames
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(Original post by Ano123)
Find the Cartesian equation of the plane  \Pi that the matrix  \mathbf{M}
 \begin{bmatrix} 2 & 1 & 0 \\ 0& 1& 0 \\ 4& 2& 0 \end{bmatrix}
maps all points to under the transformation.
Find the Cartesian equations of the two possible spheres of radii 3 that the plane  \Pi is tangent to at the origin.
[The equations should be in exact form not involving trigonometric functions.]

The last bit is aarrrd, hard to visualise.
Okay so for the first part I got the plane -2x+y=0, is that right? Been a while since FP4 now I'm trying to remember it.

The last bit is not too bad as far as visualisation goes once you have the plane equation, therefore straight forward to do.
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Ano123
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(Original post by RDKGames)
Okay so for the first part I got the plane -2x+y=0, is that right? Been a while since FP4 now I'm trying to remember it.

The last bit is not too bad as far as visualisation goes once you have the plane equation, therefore straight forward to do.
Equation of the plane is  z=2x .
What about the next part?
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RDKGames
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(Original post by Ano123)
Equation of the plane is  z=2x .
What about the next part?
Ah yeah, slipped up in my working and hadn't noticed I used y instead of z.

For the next part, you imagine that the two spheres are tangent on either side of the plane. They both have the same radius of 3 and are tangent at the origin.

So from here you can just construct a line, which is normal to plane and goes through the origin, with scalar \lambda. You can take the magnitude of this line to be 3 and solve for the two values of lambda. Sub each in and get 2 different points, these are the centres of the spheres, then proceed to put them into the sphere equation format of (x-a)^2+(y-b)^2+(z-c)^2=3^2 where the centre is (a,b,c)

That's my thinking anyway. I like the question.
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