Question
Through a point $A$ on the $x$-axis a straight line is drawn parallel to $y$-axis so as to meet the pair of straight lines $ax^{2} + 2hxy + by^{2} = 0$ in $B$ and $C$. If $AB = BC$ then :
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Solution
Sol. [Key Concept: A line through $A$(on $x$-axis) parallel to the $y$-axis meets the pair; $AB = BC$ gives the relation between $h^{2}$ and $ab$.]
Put $x = x_{1}$ in equation of pair of lines
$${\Rightarrow by^{2} + 2hx_{1}y + ax_{1}^{2} = 0^{y_{1}}\begin{matrix} & 2y_{1} \end{matrix} }{3y_{1} = - \frac{2hx_{1}}{b},2y_{1}^{2} = \frac{ax_{1}^{2}}{b} }$$
$${\Rightarrow \ \frac{9y_{1}^{2}}{2y_{1}^{2}} = \frac{4h^{2}}{b^{2}}\frac{x_{1}^{2}b}{ax_{1}^{2}} }{\Rightarrow \ \frac{9}{2} = \frac{4h^{2}}{ab}\ \Rightarrow \ 9ab = 8h^{2}}$$
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