Question

If one of the lines represented by $2x^2+2hxy+3y^2=0$ is perpendicular to one of the lines given by $3x^2+2h^{\prime }xy+2y^2=0$, then

• A. $h-h^{\prime }=0$
• B. $h+h^{\prime }=0$
• C. $3h=2h^{\prime }$
• D. $2h=3h^{\prime }$

Answer 1

The correct option is B $h+h^{\prime }=0$

Given pair of lines,

$2m^2+2h^{{}^{\prime }}m+3=0$ and ${3m}^2+2hm+2=0$ where $m=\frac{y}{x}$

Let $m_1$ and $m_2$ are roots of first equation and $m_3$ and $m_4$ are roots of second equation.

By given condition, Let $m_1\times m_3=-1$

$m_1+m_2=-h^{{}^{\prime }}$ ..............(1)

$m_1\times m_2=\frac{3}{2}$............(2)

$m_3+m_4=-h$ ..............(3)

$m_3\times m_4=\frac{2}{3}$............(4)

From (2) and (4) equations, $m_2\times m_4=-1$

Now by rewriting equation(3) by replacing terms,we get $h+h^{{}^{\prime }}=0$