perplexus.info :: Shapes : Projection Ponder

Projection Ponder (Posted on 2016-05-23)

The rays y=x and y=2x (x ≥ 0) enclose two arcs of the parabola:
y= x² + Ax+B

The two arcs are projected onto the x axis.

Let R and L denote the respective lengths of the right projection and the left projection.

Find R-L

See The Solution Submitted by K Sengupta

Rating: 5.0000 (1 votes)

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solution (if I got the algebra right)

| Comment 1 of 2

The outer two points (of the four end points) are found by

2x = x^2 + Ax + B

x^2 + (A-2)x + B = 0

The inner two points (defining the gap between the arcs) is given by

x = x^2 + Ax + B

x^2 + (A-1)x + B = 0

One arc extends from x = (2-A - sqrt((A-2)^2 - 4B)) / 2 to (1-A - sqrt((A-1)^2 - 4B)) / 2 and the other from (1-A + sqrt((A-1)^2 - 4B)) / 2 to (2-A + sqrt((A-2)^2 - 4B)) / 2

R = (2-A + sqrt((A-2)^2 - 4B)) / 2 - (1-A + sqrt((A-1)^2 - 4B)) / 2

L = (1-A - sqrt((A-1)^2 - 4B)) / 2 - (2-A - sqrt((A-2)^2 - 4B)) / 2

R - L = 2(2-A)/2 - 2(1-A)/2 = 2-A - (1-A) = 1

The answer should be 1.

Posted by Charlie on 2016-05-23 15:29:15

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