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Question
Show that the lines x2 - 4xy + y2 = 0 and the line x + y = `sqrt6` form an equilateral triangle. Find its area and perimeter.
Solution
x2 - 4xy + y2 = 0 and x + y = `sqrt6` form a triangle OAB which is equilateral.
Let OM be the perpendicular from the origin O to AB whose equation is x + y = `sqrt6`
`therefore "OM" = |(-sqrt6)/sqrt(1 + 1)| = sqrt3`
∴ area of Δ OAB = `("OM"^2)/sqrt3`
`= (sqrt3)^2/sqrt3 = sqrt3` sq.units.
In right-angled triangle OAM,
sin 60° = `"OM"/"OA"`
∴ `sqrt3/2 = sqrt3/"OA"`
∴ OA = 2
∴ length of the each side of the equilateral triangle OAB = 2 units.
∴ perimeter of Δ OAB = 3 × length of each side
= 3 × 2 = 6 units
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