Mathematics

if the two pair of lines $$2 x ^ { 2 } + 6 x y + y ^ { 2 } = 0$$ and $$4 x ^ { 2 } + 18 x y + b y ^ { 2 } = 0$$ are equally inclined,then $$b$$ is equal to


ANSWER

$$1$$


SOLUTION
Given pair of lines are

$$2x^2 +6xy+y^2=0$$...(1)

and $$ 4x^2+18xy +by^2=0$$...(2)

eq. of angle bisector of eq. (1)

$$\dfrac {x^2-y^2}{2-1}=\dfrac {xy}{3}$$...(3)

eq. of angle bisector of eq. (2)

$$\dfrac {x^2-y^2}{2-1}=\dfrac {xy}{9}$$...(4)

from eq (3)& (4)

$$\dfrac {x^2-y^2}{xy}=\dfrac {1}{3}=\dfrac {4-b}{9}$$

$$\Rightarrow \dfrac {4-b}{9}=\dfrac {1}{3}$$

$$\Rightarrow 4-b =3$$

$$\Rightarrow b=1$$
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