Đáp án:
$\begin{array}{l}
a)\dfrac{x}{3} = \dfrac{y}{4} = k\\
\Leftrightarrow \left\{ \begin{array}{l}
x = 3k\\
y = 4k
\end{array} \right.\\
\Leftrightarrow {x^2} + {y^2} = {\left( {3k} \right)^2} + {\left( {4k} \right)^2} = 100\\
\Leftrightarrow 25{k^2} = 100\\
\Leftrightarrow {k^2} = 4\\
\Leftrightarrow \left[ \begin{array}{l}
k = 2\\
k = - 2
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = 6;y = 8\\
x = - 6;y = - 8
\end{array} \right.\\
b)\dfrac{x}{4} = \dfrac{y}{3} = k \Leftrightarrow \left\{ \begin{array}{l}
x = 4k\\
y = 3k
\end{array} \right.\\
\Leftrightarrow x.y = 300\\
\Leftrightarrow 4k.3k = 300\\
\Leftrightarrow 12{k^2} = 300\\
\Leftrightarrow {k^2} = 25\\
\Leftrightarrow \left[ \begin{array}{l}
k = 5 \Leftrightarrow x = 20;y = 15\\
k = - 5 \Leftrightarrow x = - 20;y = - 15
\end{array} \right.\\
c)\dfrac{x}{5} = \dfrac{y}{3} = k \Leftrightarrow \left\{ \begin{array}{l}
x = 5k\\
y = 3k
\end{array} \right.\\
{x^2} - {y^2} = 16\\
\Leftrightarrow {\left( {5k} \right)^2} - {\left( {3k} \right)^2} = 16\\
\Leftrightarrow 16{k^2} = 16\\
\Leftrightarrow {k^2} = 1\\
\Leftrightarrow \left[ \begin{array}{l}
k = 1 \Leftrightarrow x = 5;y = 3\\
k = - 1 \Leftrightarrow x = - 5;y = - 3
\end{array} \right.\\
d)\dfrac{x}{2} = \dfrac{y}{5} = k \Leftrightarrow \left\{ \begin{array}{l}
x = 2k\\
y = 5k
\end{array} \right.\\
2{x^2} + 3{y^2} = 332\\
\Leftrightarrow 2.{\left( {2k} \right)^2} + 3.{\left( {5k} \right)^2} = 332\\
\Leftrightarrow 83{k^2} = 332\\
\Leftrightarrow {k^2} = 4\\
\Leftrightarrow \left[ \begin{array}{l}
k = 2 \Leftrightarrow x = 4;y = 10\\
k = - 2 \Leftrightarrow x = - 4;y = - 10
\end{array} \right.\\
e)\dfrac{x}{3} = \dfrac{y}{4} = k \Leftrightarrow \left\{ \begin{array}{l}
x = 3k\\
y = 4k
\end{array} \right.\\
{x^3} + {y^3} = 91\\
\Leftrightarrow {\left( {3k} \right)^3} + {\left( {4k} \right)^3} = 91\\
\Leftrightarrow 91{k^3} = 91\\
\Leftrightarrow k = 1\\
\Leftrightarrow x = 3;y = 4\\
e)\dfrac{x}{5} = \dfrac{y}{4} = k \Leftrightarrow \left\{ \begin{array}{l}
x = 5k\\
y = 4k
\end{array} \right.\\
{x^2}y = 100\\
\Leftrightarrow {\left( {5k} \right)^2}.4k = 100\\
\Leftrightarrow 100{k^3} = 100\\
\Leftrightarrow k = 1\\
\Leftrightarrow x = 5;y = 4
\end{array}$
