If the slopes of the lines given by the equation $a x^{2}+2 h x y+b y^{2}=0 \quad$ are in the ratio $5: 3$, then the ratio $h^{2}: a b=$

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Aicharya · Accepted Answer

Let $y=m_{1} x$ and $y=m_{2} x$ be the lines represented by the equation. $a x^{2}+2 h x y+b y^{2}=0$ Then, $m_{1}+m_{2}=\frac{-2 h}{b}$ and $m_{1} m_{2}=\frac{a}{b}$ We have, $\frac{m_{1}}{m_{2}}=\frac{5}{3} \Rightarrow m_{1} \Rightarrow \frac{5 m_{2}}{3}$ $	herefore \frac{5 \mathrm{~m}_{2}}{3}+\mathrm{m}_{2}=\frac{-2 \mathrm{~h}}{\mathrm{~b}}$ and $\left(\frac{5 \mathrm{~m}_{2}}{3}ight) \mathrm{m}_{2}=\frac{\mathrm{a}}{\mathrm{b}}$ $	herefore \frac{8 m_{2}}{3}=\frac{-2 h}{b} \Rightarrow m_{2}=\frac{-3 h}{4 b}$ and $m_{2}^{2}=\frac{3 a}{5 b}$ $\left(\frac{-3 h}{4 b}ight)^{2}=\frac{3 a}{5 b} \Rightarrow \frac{9 h^{2}}{16 b^{2}}=\frac{3 a}{5 b}$ $	herefore \frac{\mathrm{h}^{2}}{\mathrm{ab}}=\frac{16}{15}$

If the slopes of the lines given by the equation $a x^{2}+2 h x y+b y^{2}=0 \quad$ are in the ratio $5: 3$,…

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