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If the slope of one of the lines given by $a x^2+2 h x y+b y^2=0$ is two times the other, then
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$8 h^2=9 a b$
$\begin{aligned} & \mathrm{m}_1+\mathrm{m}_2=\frac{-2 \mathrm{~h}}{\mathrm{~b}} \text { and } \mathrm{m}_1+\mathrm{m}_2=\frac{\mathrm{a}}{\mathrm{b}} \\ & \Rightarrow \mathrm{m}+2 \mathrm{~m}=\frac{-2 \mathrm{~h}}{\mathrm{~b}} \text { and } \mathrm{m} \times 2 \mathrm{~m}=\frac{\mathrm{a}}{\mathrm{b}} \\ & \Rightarrow 3 \mathrm{~m}=\frac{-2 \mathrm{~h}}{\mathrm{~b}} \text { and } 2 \mathrm{~m}^2=\frac{\mathrm{a}}{\mathrm{b}} \\ & \Rightarrow 2\left(\frac{-2 \mathrm{~h}}{3 \mathrm{~b}}\right)^2=\frac{\mathrm{a}}{\mathrm{b}} \\ & \Rightarrow 8 \mathrm{~h}^2=9 \mathrm{ab}\end{aligned}$
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