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The distance between the lines represented by the equation x 2 +2root2 xy+2y 2 +4x+4root2 y+1 is:

The distance between the lines represented by the equation x2+2root2 xy+2y2+4x+4root2 y+1 is:
 

Grade:11

1 Answers

Riddhish Bhalodia
askIITians Faculty 434 Points
8 years ago
f(x,y) = x^2 + 2\sqrt{2}xy + 2y^2 + 4x + 4\sqrt{2}y + 1 = 0
f(x,y) = (x + \sqrt{2}y)^2 + 4(x + \sqrt{2}y) + 1 = 0
let t = (x + \sqrt{2}y)
then
t^2 + 4t +1 =0
which yields
t = -2+\sqrt{3}, \quad -2-\sqrt{3}
hence the two lines are as follows
x + \sqrt{2}y = -2+\sqrt{3} \quad x + \sqrt{2}y = -2-\sqrt{3}
these are parallel lines
When the lines are given by
ax+by+c1=0
ax+by+c2=0
the distance between them can be expressed as
d = \frac{|c_2-c_1|}{\sqrt {a^2+b^2}}
so in this case
d = \frac{2\sqrt{3}}{\sqrt {3}} = 2


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