An astronomical researcher is using software to study the parabolic path of an asteroid. By entering the equation for the path of the asteroid, the researcher could determine if the asteroid would collide with any other bodies in the near future. The equation for the asteroid’s path is x^2-2xy+y^2-2(√2)x-2(√2)y=0. However, the application cannot accept the xy term; instead, the software allows the researcher to rotate the parabola. If the coordinate axes are rotated to coincide with the axes of the parabola, the researcher can use the equation of the new parabola and provide the angle of rotation. What will be the equation of the parabola on the rotated axes?
A) x'^2 = -16y'
B) x'^2 = 4y'
C) y'^2 = -4x'
D) y'^2 = 8x'
E) y'^2 = 2x'
general equation of a conic:
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
we can eliminate the xy term by finding the angle of rotation Ø , so that
tan (2Ø) = B/(A-C)
and replacing the x and y with a new X and Y, so that
x = XcosØ - YsinØ and y = XsinØ + YcosØ
For our equation: A = 1 , B = -2 , C = 1
tan 2Ø = -2(1-1) = -2/0 which is undefined
but I know tan 90° is undefined
so 2Ø = 90°
Ø = 45° , and of course both sin45 and cos45 = √2/2
so x = √2/2 X - √2/2 Y and y = √2/2 X + √2/2 Y
x = (√2/2)(X-Y)
y = (√2/2)(X+Y)
sub these back into the original equation, which I will leave up to you, will yield the equation with the x-axis and y-axis as the proper orientation.
If you have time here is a 1 1/2 hour video of the whole topic
https://www.youtube.com/watch?v=vIFSMRpmpRo
(you could of course advise the researcher to get more uptodate software to graph the equation as is.
Even Wolfram, the best free software of this type, can do it
http://www.wolframalpha.com/input/?i=x%5E2-2xy%2By%5E2-2%28√2%29x-2%28√2%29y%3D0+%2C
To find the equation of the parabola on the rotated axes, we need to perform a rotation of coordinates. Let's go through the steps:
Step 1: Rewrite the original equation in the general form of a second-degree equation (ax^2 + bxy + cy^2 + dx + ey + f = 0).
The given equation is x^2 - 2xy + y^2 - 2√2x - 2√2y = 0.
Step 2: Identify the coefficients of all the variables in the general form equation.
From the given equation, we have:
a = 1, b = -2, c = 1, d = -2√2, e = -2√2, f = 0.
Step 3: Calculate the angle of rotation (θ) using the formula:
tan(2θ) = b / (a - c)
Here, b = -2 and a - c = 1 - 1 = 0. So the equation becomes:
tan(2θ) = -∞ (since 0 divided by any non-zero number is infinity)
Therefore, 2θ = π/2 or 3π/2.
Step 4: Determine the equation of the parabola on the rotated axes.
For the equation of a parabola oriented with the rotated axes, the xy-term should be eliminated.
By substituting x = x'cosθ - y'sinθ and y = x'sinθ + y'cosθ, the general form equation can be rewritten as:
(x'cosθ - y'sinθ)^2 - 2(x'cosθ - y'sinθ)(x'sinθ + y'cosθ) + (x'sinθ + y'cosθ)^2 - 2√2(x'cosθ - y'sinθ) - 2√2(x'sinθ + y'cosθ) = 0.
Expanding and simplifying, we get:
(x'^2cos^2θ - 2x'y'sinθcosθ + y'^2sin^2θ) - 2(x'^2sinθcosθ + y'^2sin^2θ + x'y'cos^2θ - x'y'sin^2θ) - 2√2(x'cosθ - y'sinθ) - 2√2(x'sinθ + y'cosθ) = 0.
Rearranging the terms and applying trigonometric identities (cos^2θ + sin^2θ = 1), we obtain:
(x'^2 - 2√2cosθ(x' + y') + y'^2 - 2√2sinθ(x' - y')) = 0.
Comparing this equation with the general form, we can determine the coefficients for the rotated axes equation:
a' = 1, b' = -2√2cosθ, c' = 1, d' = -2√2sinθ, e' = -2√2cosθ, f' = 0.
Step 5: Substitute the values of cosθ and sinθ based on the possible values of 2θ (π/2 or 3π/2).
For 2θ = π/2, cosθ = 0 and sinθ = 1. Plugging these values into the coefficients, we get:
a' = 1, b' = 0, c' = 1, d' = -2√2, e' = 0, f' = 0.
Therefore, the equation on the rotated axes is: x'^2 - 2√2y' = 0.
For 2θ = 3π/2, cosθ = 0 and sinθ = -1. Plugging these values into the coefficients, we get the same equation as above:
x'^2 - 2√2y' = 0.
So the equation of the parabola on the rotated axes is x'^2 - 2√2y' = 0.
Therefore, the correct answer is option (A) x'^2 = -16y'.