Question

If the two pairs of lines $2x^2+6xy+y^2=0$ and $4x^2+18xy+b{y}^2=0$ are equally inclined then $b=$

• A. $1$
• B. $-1$
• C. $2$
• D. $-2$

Answer 1

The correct option is A $1$

The pair of lines $2x^2+6xy+y^2=0$ will be equally inclined to the pair $4x^2+18xy+b{y}^2=0$ if the bisectors of the angles between $2x^2+6xy+y^2=0$ are the same as the bisectors of the angle between $4x^2+18xy+b{y}^2=0$.

The equation of the bisectors of the angles between the line pair $2x^2+6xy+y^2=0$ is,

$\frac{x^2-y^2}{a-b}=\frac{xy}{h}$

$\frac{x^2-y^2}{2-1}=\frac{xy}{3}$

$\frac{x^2-y^2}{xy}=\frac{1}{3}$ ------ (i)

The equation of the bisectors of the angles between the line pair $4x^2+18xy+b{y}^2=0$ is,

$\frac{x^2-y^2}{a-b}=\frac{xy}{h}$

$\frac{x^2-y^2}{4-b}=\frac{xy}{9}$

$\frac{x^2-y^2}{xy}=\frac{4-b}{9}$------ (ii)

From (i) and (ii)

$\frac{1}{3}=\frac{4-b}{9}$

$9=12-3b$

$3b=3$

$b=\frac{3}{3}$

$b=1$