SOLUTION: Length of side of an equilateral triangle inscribed in a parabola y^2 -2x-2y-3=0 whose one angular point is vertex of the parabola

Question 1011076: Length of side of an equilateral triangle inscribed in a parabola y^2 -2x-2y-3=0 whose one angular point is vertex of the parabola
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
Length of side of an equilateral triangle inscribed in a parabola y^2 -2x-2y-3=0 whose one angular point is vertex of the parabola
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The vertex is (-2,1)
The LOS of the parabola is y = 1
---
Shift the parabola down 1 unit and right 2 units to the vertex is (0,0)
--> y^2 = 2x
================
The angles of the sides of the vertex of the triangle are +30 degs and -30 degs wrt the x-axis.
----
m1 = atan(30) = sqrt(3)/3
m2 = -sqrt(3)/3
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Equation of line with slope m1 thru (0,0) is
y = m1*x = x*sqrt(3)/3
x = y*sqrt(3)
Sub for x in y^2 = 2x
y^2 = 2y*sqrt(3)
----
y^2 -2sqrt(3)y = 0
--> intersections at y = 0 (the vertex)
and at y = 2sqrt(3)
--> 1/2 side length = 2sqrt(3)
sides = 4sqrt(3)

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