<s>[INST] Rotation and translation of coordinate axes I am studying rotation and translation of conical but have no doubt in basic concept (Sorry, I know this is a very stupid question but I'm really struggling to understand). Especially in this equation:
$$
9x^2 - 4y^2 - 18x - 16y - 7 = 0
$$
$$
\begin{cases}
x = u + h\\
y = v + k
\end{cases}\\
9u^2 - 4v^2 + (18h - 18)u + (-8k - 16)v + G(h, k) = 0 
$$
Why is not this equation?
$$
9u^2 - 4v^2 - 18(u + h) - 16(v + k) + G(h, k) = 0 
$$
 [/INST] What happens when you substitute $u+h$ for $x$ in the equation
$9x^2 - 4y^2 - 18x - 16y - 7 = 0$?
First of all, $x^2 = (u + h)^2 = u^2 + 2hu + h^2$.
Therefore $$9x^2 = 9(u^2 + 2hu + h^2) = 9u^2 + 18hu + 9h^2.$$
And of course $-18x = -18u - 18h$. So
\begin{align}
9x^2 - 4y^2 - 18x - 16y - 7
 & = 9u^2 + 18hu + 9h^2 - 18u - 18h - 4y^2 - 16y - 7 \\
 & = 9u^2 + (18h - 18)u - 4y^2 - 16y + 9h^2 - 18h - 7.
\end{align}
Now substitute $v+k$ for $y$.
You should find that $-4y^2 = -4(v+k)^2 = -4v^2 - 8kv - 4k^2$
and $-16y = -16v - 16k$. The result is
\begin{align}
9x^2 - & 4y^2 - 18x - 16y - 7 \\
 & = 9u^2 + (18h - 18)u - 4v^2 - 8kv - 4k^2 - 16v - 16k + 9h^2 - 18h - 7 \\
 & = 9u^2 + (18h - 18)u - 4v^2 + (-8k - 16)v + (- 4k^2 - 16k + 9h^2 - 18h - 7).
\end{align}
Now, of the two equations that you wrote, which one could possibly be
equal to $9x^2 - 4y^2 - 18x - 16y - 7$?
As a hint, where is the $hu$ term in the second equation?
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