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? $$ 9u^2 - 4v^2 - 18(u + h) - 16(v + k) + G(h, k) = 0 $$
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?