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Changes in equation coefficients produced by translation. What changes are produced in the coefficients of the general equation of the second degree

             f(x, y) = ax2 + 2bxy + cy2 + 2dx + 2ey + f = 0

by a translation of the coordinate system? Let us observe the results of translating the coordinate system to the point (x0, y0). Let x'-y' be a coordinate system with origin located at point (x0, y0) of the x-y system. We substitute

x = x' + x0

y = y' + y0

into equation 1) above.

The equation becomes

f(x' + x0, y' + y0) = a(x' + x0)2 + 2b(x' + x0)(y' + y0) + c(y' + y0)2 + 2d(x' + x0) + 2e (y' + y0) + f


             = ax' 2 + 2bx'y' + cy' 2 + 2(ax0 + by0 + d)x' + 2(bx0 + cy0 + e)y' + ax02 + 2bx0y0 + cy02 + 2dx0 + 2ey0 + f


            = ax' 2 + 2bx'y' + cy' 2 + 2(ax0 + by0 + d)x' + 2(bx0 + cy0 + e)y' + f(x0, y0) = 0


We see that a translation has no effect on the coefficients of the second degree terms (i.e. the coefficients a, b, c). It only changes the first degree coefficients d and e and the constant term f where


Changes in equation coefficients produced by rotation. Under a rotation of the coordinate system about its origin by an angle of θ degrees the general equation of the second degree


            f(x, y) = ax2 + 2bxy + cy2 + 2dx + 2ey + f = 0


            a'x' 2 + 2b'x'y' + c'y'2 + 2d'x' + 2e'y' + f ' = 0


            a' = a cos2 θ + 2 b sin θ cos θ + c sin2 θ

            b' = (c - a) sin θ cos θ + b (cos2 θ - sin2 θ)

                        = ½ (c - a) sin 2θ + b cos 2θ

            c' = a sin2 θ - 2b sin θ cos θ + c cos2 θ

            d' = d cos θ + e sin θ

            e' = e cos θ - d sin θ

            f ' = f


where the equations of the transformation are

x = x' cos θ - y' sin θ

y = x' sin θ + y' cos θ .

We note that all of the coefficients of the equation change under the rotation of the coordinate system. Only the constant term f remains unchanged.

Invariance of the quantity b2 - ac under rotation. The quantity b2 - ac is invariant under rotation of coordinates. In other words, if the general equation of the second degree

            f(x, y) = ax2 + 2bxy + cy2 + 2dx + 2ey + f = 0

is transformed by

            x = x' cos θ - y' sin θ

            y = x' sin θ + y' cos θ

to obtain

            a'x' 2 + 2b'x'y' + c'y'2 + 2d'x' + 2e'y' + f ' = 0


            b' 2 - a'c' = b2 - ac .

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