MHB Find the equation of the conicoid

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To find the equation of the conicoid given by 2x² - y² = z² + 2x - 7 after shifting the origin to (2, -2, 0) and rotating the axes, one must first substitute the new coordinates into the equation. The transformation involves adjusting the variables to account for the new origin and applying the rotation matrix corresponding to the specified direction ratios. The discussion emphasizes the importance of showing prior work to facilitate better assistance from others. This approach ensures that helpers can provide relevant guidance without repeating suggestions that have already been attempted. The goal is to derive the new equation of the conicoid in the transformed coordinate system.
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Find the equation of the conicoid
$$2x^2-y^2=z^2+2x-7$$
when the origin is shifted to
$$(2,-2,0)$$
and the axes are rotated so that the new axes have direction ratios
$$-1,0,1;1,-2,1;0,1,1$$
 
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Hello debrajr and welcome to MHB! :D

We ask that our users show their progress (work thus far or thoughts on how to begin) when posting questions. This way our helpers can see where you are stuck or may be going astray and will be able to post the best help possible without potentially making a suggestion which you have already tried, which would waste your time and that of the helper.

Can you post what you have done so far?
 

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