Eliminate $p$ from these 2 equations.

Question

$$ X \ = \ 2 \left[ \dfrac {h_1pv_1} {(1-p^2v_1^2)^{1/2}} + \dfrac {h_2pv_2} {(1-p^2v_2^2)^{1/2}} \right] \\ T_2 \ = \ 2 \left[ \dfrac {h_1/v_1} {(1-p^2v_1^2)^{1/2}} + \dfrac {h_2/v_2} {(1-p^2v_2^2)^{1/2}} \right] $$

The Objective here is to eliminate $p$ from both these equations.

Answer 1

See below the first steps to start:

$$\left\{{\begin{align*} pv_1^2T_2-X&=\dfrac {h_2 p \left( \dfrac {v_1^2-v_2^2} {v_2} \right) } {(1-p^2v_2)^{1/2}} \\ \ \\ pv_2^2T_2-X&=\dfrac {h_1 p \left( \dfrac {v_2^2-v_1^2} {v_1} \right) } {(1-p^2v_1)^{1/2}} \end{align*}}\right. \\ \ \ \\ \ \ \\ \dfrac {pv_1^2T_2-X} {pv_2^2T_2-X} = -\dfrac {h_2v_1} {h_1v_2} \left(\dfrac {1-p^2v_1} {1-p^2v_2} \right)^{1/2} \ \to \ p^2=\ldots?$$