If slope of one of the lines $a x^2+2 h x y+b y^2=0$ is twice that of the other, then $\mathrm{h}^2: \mathrm{ab}$ is

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Aicharya · Accepted Answer

$a x^2+2 h x y+b y^2=0$ We have $\mathrm{m}_1+\mathrm{m}_2=\frac{-2 \mathrm{~h}}{\mathrm{~b}}$ and $\mathrm{m}_1 \mathrm{~m}_2=\frac{\mathrm{a}}{\mathrm{b}}$ $\begin{aligned} & 	ext { Also } \mathrm{m}_1+2 \mathrm{~m}_2 \ & 	herefore 3 \mathrm{~m}_2=-\frac{2 \mathrm{~h}}{\mathrm{~b}} \Rightarrow \mathrm{m}_2=\frac{-2 \mathrm{~h}}{3 \mathrm{~b}} \quad 	ext { and } 2 \mathrm{~m}_2^2=\frac{\mathrm{a}}{\mathrm{b}} \ & 	herefore 2\left(\frac{-2 \mathrm{~h}}{3 \mathrm{~b}}ight)^2=\frac{\mathrm{a}}{\mathrm{b}} \Rightarrow \frac{8 \mathrm{~h}^2}{9 \mathrm{~b}^2}=\frac{\mathrm{a}}{\mathrm{b}} \Rightarrow \frac{\mathrm{h}^2}{\mathrm{ab}}=\frac{9}{8} \end{aligned}$

If slope of one of the lines $a x^2+2 h x y+b y^2=0$ is twice that of the other, then $\mathrm{h}^2:…

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