If the slope of one of the lines represented by ax^+2hxy+by^=0 is thre

1.	If the slope of one of the lines represented by ax^+2hxy+by^=0 is three times the other then prove that 3h^=4ab
Answer» Geometry Given: the slope of ONE of the LINES represented by $ax^{2}+2hxy+by^{2}=0$ is three TIMES the other To PROVE: $3h^{2}=4ab$ Proof: Let $y^{2}+\frac{2h}{b}xy+\frac{a}{b}x^{2}=(y-m_{1}x)(y-m_{2}x)$ Then, $\quad m_{1}+m_{2}=-\frac{2h}{b}$ $\quad m_{1}m_{2}=\frac{a}{b}$ Given, $m_{1}=3m_{2}\Rightarrow \frac{m_{1}}{m_{2}}=3$ Now, $\frac{(m_{1}+m_{2})^{2}}{m_{1}m_{2}}=\frac{(-2h/b)^{2}}{a/b}$ $\Rightarrow \frac{m_{1}^{2}+2m_{1}m_{2}+m_{2}^{2}}{m_{1}m_{2}}=\frac{4h^{2}/b^{2}}{a/b}$ $\Rightarrow \frac{m_{1}}{m_{2}}+2+\frac{1}{\frac{m_{1}}{m_{2}}}=\frac{4h^{2}}{ab}$ $\Rightarrow 3+2+\frac{1}{3}=\frac{4h^{2}}{ab}$ $\Rightarrow \frac{4h^{2}}{ab}=\frac{16}{3}$ $\Rightarrow\frac{h^{2}}{ab}=\frac{4}{3}$ $\Rightarrow 3h^{2}=4ab$ This completes the proof.