Why does an orthogonal linear transformation does not distort conics while other transformations do?

Question

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Why does the orthogonal transformation preserve a conic (send it to a congruent conic), but other transformations only preserve the general shape of the conic (e.g send hyperbolas to hyperbolas but not congruent hyperbolas)?

Answer 1

$$ |v|^2=v^Tv;\quad |v'|^2=v'^Tv'=(Ov)^T(Ov)=v^TO^TOv. $$ Hence $|v|^2=|v'|^2$ if and only if $O^TO=I$.