Question

Find p and q, if the following equation represents a pair of perpendicular lines
2x²+ 4xy – py² + 4x +qy+1=0​

Answer 1

$\text{Condition for the pair of straight lines ax^2+2hxy+by^2+2gx+2fy+c=0 }$

$\text{to be perpendicular is a+b=0}$

$\text{Given equation is }$

$2x^2+ 4xy-py^2+4x+qy+1=0$

$\text{Here,}$​

$\text{a=2, h=2, b=-p, g=2, f=q/2, c=1}$

$\text{since the lines are perpendicuar, a+b=0}$

$\implies\;2-p=0$

$\implies\bf\,p=2$

$\text{Condition for pair of straight lines is }\bf\;abc+2fgh-af^2-bg^2-ch^2=0$

$(2)(-2)(1)+2(q/2)(2)(2)-2(q^2/4)+2(4)-1(4)=0$

$-4+4q-q^2/2+4=0$

$4q-q^2/2=0$

$8q-q^2=0$

$q(8-q)=0$

$\bf\;q=0\;\text{or}\;q=8$