Prove that the straight lines $a x^{2} + 2 h x y + b y^{2} = 0$ make equal angles with the axis of $x$ if $h = a cos ω$ , the axes being inclined at an angle $ω$ .

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Solution

Let the equations represented by $a x^{2} + 2 h x y + b y^{2} = 0$ be $y - m_{1} x = 0$ and $y - m_{2} x = 0$ then
$m_{1} + m_{1} = - \frac{2 h}{b}, m_{1} m_{2} = \frac{a}{b}$
If the lines males equal angles with x axis, one angle is $θ$ and the other is $180 - θ$
$tan θ = \frac{m_{1} sin ω}{1 + m_{1} cos ω} . . . .1$
and $tan (180 - θ) = \frac{m_{2} sin ω}{1 + m_{2} cos ω} . . . .2$
By $1$ and $2$ we get
$\frac{m_{1} sin ω}{1 + m_{1} cos ω} = - \frac{m_{2} sin ω}{1 + m_{2} cos ω}$
as $tan (180 - θ) = - tan θ$
Putting the values of $m_{1} + m_{2}$ and $m_{1} m_{2}$
$h = a cos ω$

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