1. Show that the coordinates of point (x,y) rotated counter-clockwise by an angle of θ around the origin have the new coordinates (xcosθ - ysinθ, xsinθ + ycosθ). Draw a picture that clearly illustrates your derivation, and provide plenty of explanation. Hint: consider the distance from the origin to (x,y) as "d", and use the angle sum formulas.

2. Use this result to find the equation of the hyperbola xy = 2 rotated 45[SIZE=-1]° counterclockwise. Compute the vertices, foci, and asymptotes of each of these hyperbolas.

3. Derive the equation of the general conic

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

when rotated counterclockwise through an angle [/SIZE]θ. Hint: the coefficient of x^2 becomes (A cos^2θ + Bsinθcosθ + Csin^2θ)

4. Show that rotating the equation through an angle of θ where tan 2θ = B/(A-C) will eliminate that pesky xy term.

5. For each of the following, find:

-the angle of rotation that eliminates the xy term

-the equation resulting from that rotation

-the vertices, foci, and, if appropriate, asymptotes of each equation

a. x^2 + 4xy + y^2 - 3 = 0

b. 3x^2 - 10xy + 3y^2 - 32 = 0

c. x^2 + 4xy + 4y^2 + 5(root5y) + 5 = 0

I know it's lengthy and pretty annoying, but any help at all, no matter how little, is greatly appreciated. Thanks in advance!