Consider the following xy 9 Use the discriminant to determi

Consider the following, xy = 9 Use the discriminant to determine whether the graph of the equation is a parabola, an ellipse, or a hyperbola. parabola ellipse hyperbola. Use a rotation of axes to eliminate the xy-term. (Write an equation in XY-coordinates. Use a rotation angle that satisfies 0 lessthanorequalto phi lessthanorequalto pi/2.)

Solution

Solution:(b)

Given Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

x = x\' cos() - y\' sin()
y = x\' sin() + y\' cos()

where cot(2) = (A-C)/B

eliminates the xy term.

In this case, A = C = 0, so cot(2) = 0, which means cos(2) = 0. One solution is = /4. So

x = (1/2)x\' - (1/2)y\'
y = (1/2)x\' + (1/2)y\'

xy = {(1/2)x\' - (1/2)y\'} * {(1/2)x\' + (1/2)y\'}

    = [(1/2)x\']^2 + [(1/2)x\'] [(1/2)y\'] - [(1/2)x\'] [(1/2)y\'] - [(1/2)y\']²

    = (1/2)(x\'^2 - y\'^2)

Therefore, rotation by /4 transforms xy = 9 to (1/2)(x\'^2 - y\'^2) = 9. Written in standard form,

x\'^2/(32)^2 - y\'^2/(32)^2 = 1