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Points A and B are on parabola y=3x^2-5x-3, and the origin is the midpoint of AB. Find the square of the length of
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AB.
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AB means line AB
y = 3x^2 - 5x - 3
Let A = (u, 3u^2 - 5u -3)
Let B = (v, 3v^2 - 5v - 3)
Using the midpoint, rule, we have that
[ u + v ] / 2 = 0 and [ 3u^2 - 5u - 3 + 3v^2 - 5v - 3 ] / 2 = 0 (2)
u + v = 0
v = -u (1)
Sub (1) inot (2) and we have that
[3u^2 - 5u - 3 + 3 (-u)2 - 5(-u) - 3 ] / 2 = 0 multiply through by 2 and simplify
6u^2 - 6 = 0 divide through by 6
u^2 - 1 = 0 factor
(u + 1) ( u -1) = 0
Set both factors to 0 and solve for u ⇒ u = -1 or u = 1
If u = -1 then the associated y coordinate is 3(1)^2 - 5(-1) - 3 = 5
So A = (-1, 5)
And the x coordinate of B is -u = 1 and the y coordinate of B is 3(1)^2 - 5(1) - 3 = -5
So, in this case B = (1, -5)
The slope of this line through A and B is [ -5 - 5] / [ 1 - -1] = -10/2 = -5
And the equation of this line is
y = -5 (x -1) - 5
y = -5x
So....the distance between A and B is √ [ -5 - 5)^2 + ( -1 -1)^2 ] = √[ 10^2 + 2^2] = √104
So....the square of the length of AB is just 104
Note that if u = 1, we have the same results...A and B are just "switched"
See the graph, here : https://www.desmos.com/calculator/e2uvdwwhl3