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Analytical Geometry> The distance between the lines represente...

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

Amogh Dasture , 9 Years ago

Grade 11

1 Answers

Riddhish Bhalodia

Last Activity: 9 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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