The equation 2 x^2-4 x y- p y^2+4 x+ q y+1=0 will represent two m… - Vidyadip

MCQ

The equation $2 x^2-4 x y- p y^2+4 x+ q y+1=0$ will represent two mutually perpendicular straight lines, if

A
$p=1$ and $q=2$ or 6
B
$p=2$ and $q=0$ or 6
✓
$p =2$ and $q =0$ or 8
D
$p=-2$ and $q=-2$ or 8

✓

Answer

Correct option: C.

$p =2$ and $q =0$ or 8

(C) Given equation of pair of lines is $2 x^2-4 x y- p y^2+4 x+ q y+1=0$
$a =2, b=- p , c =1, f =\frac{ q }{2}, g=2, h=-2$
The lines are perpendicular,
$\therefore \quad a+b=0$
$\Rightarrow 2-p=0 \Rightarrow p=2$
The equations represents pair of lines
$\therefore \quad 2(-2)(1)+2\left(\frac{q}{2}\right)(2)(2)-2\left(\frac{q}{2}\right)^2$ $+2(2)^2-1(2)^2=0$
$\Rightarrow q^2-8 q=0 \Rightarrow q=0$ or 8

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