Question

If the pair of line given by $a{x}^2+2hxy+b{y}^2=0$ are rotated through origin about ${90}^{\circ }$, then find combined equation of their equation in the new position.

Answer 1

Let the lines be represented by the equation $a{x}^2+2hxy+b{y}^2=0$ be $y=m_{1}x$.....(1) and $y=m_{2}x$.....(2).

Where, $m_1+m_2=-\frac{2h}{b}$.....(3) and $m_1.m_2=\frac{a}{b}$......(4)

Now the lines perpendicular to the lines (1) and (2) respectively be,

$y=-\frac{x}{m_1}$ and $y=-\frac{x}{m_2}$.

So the pair of lines represented by them is,

$(y+\frac{x}{m_1})(y+\frac{x}{m_2})=0$

or, $y^2+\frac{m_1+m_2}{m_1.m_2}xy+\frac{x^2}{m_1.m_2}=0$

or, $a{y}^2-2hxy+b{x}^2=0$ [ Using (3) and (4)]