Examples on Pairs of Lines Through the origin Set 2
Example - 17
Find the equation of the pair of lines through the origin and perpendicular to the pair of lines \(a{x^2} + 2hxy + b{y^2} = 0.\)
Solution: Let the slopes of the two lines represented by the given equation be \(m_1\) and \(m_2.\) As explained earlier, \(m_1\) and \(m_2\) are the roots of the quadratic
\[b{m^2} + 2hm + a = 0\]
so that
\[\begin{align}{m_1} + {m_2} = \frac{{ - 2h}}{b}, \qquad {m_1}{m_2} = \frac{a}{b} \qquad \qquad ...(1)\end{align}\]
The slopes of the lines whose joint equation we require will simply be \(\begin{align}\frac{{ - 1}}{{{m_1}}}\;{\rm{and}}\;\frac{{ - 1}}{{{m_2}}}\end{align}\) so that their equations will be:
\[\begin{align} \qquad& y = \frac{{ - 1}}{{{m_1}}}x, \qquad \qquad \qquad y = \frac{{ - 1}}{{{m_2}}}x\\ \Rightarrow \qquad& x + {m_1}y = 0, \qquad \quad \quad \;\;\, x + {m_2}y = 0 \end{align}\]
The required joint equation is
\[\begin{align} \qquad & (x + {m_1}y)(x + {m_2}y) = 0\\ \Rightarrow \qquad & {x^2} + {m_1}{m_2}{y^2} + ({m_1} + {m_2})xy = 0\\ \Rightarrow \qquad & {x^2} + \frac{a}{b}{y^2} - \frac{{2h}}{b}xy = 0 \qquad \qquad \qquad \qquad \qquad (Using\;(1))\\ \Rightarrow \qquad & b{x^2} - 2hxy + a{y^2} = 0 \end{align}\]
Example - 18
The equation \(a{x^3} + b{x^2}y + c{x^2} + d{y^3} = 0\) is a third degree homogenous equation and hence represents three straight lines passing through the origin. Find the condition so that two of these three lines may be perpendicular.
Solution: We divide the given equation by \(x^3\) and substitute \(\begin{align}\frac{y}{x}=m\end{align}\) to obtain:
\[d{m^3} + c{m^2} + bm + a = 0 \qquad \qquad \qquad ...(1)\]
This has three roots, say \({m_1},\,{m_2},\,{m_3},\) corresponding to the three straight lines. Since we want two of these lines to be perpendicular, we can assume
\[m_1m_2=-1\]
From (1), we have
\[\begin{align} &{m_1}\,{m_2}\,{m_3} = \frac{{ - a}}{d}\\ \Rightarrow \qquad & {m_3} = \frac{a}{d} \end{align}\]
Substituting this value of \(m_3\) back in (1), (since \(m_3\) is a root of (1)), we obtain
\[\begin{align} & \frac{{d{a^3}}}{{{d^3}}} + \frac{{c{a^2}}}{{{d^2}}} + \frac{{ba}}{d} + a = 0\\ \Rightarrow \qquad & {a^2} + ac + bd + {d^2} = 0 \end{align}\]