Rotating Conics (10.4.4)

Question 1

• The general equation of a conic with axes parallel to the xy-axes is Ax2 + Cy2 + Dx + Ey + F = 0. These types of conics can be graphed by writing the equation in the standard form of a parabola, ellipse, circle, or hyperbola, and then graphing as usual.

Answer 1

• The general equation of a conic with axes parallel to the xy-axes is Ax2 + Cy2 + Dx + Ey + F = 0. These types of conics can be graphed by writing the equation in the standard form of a parabola, ellipse, circle, or hyperbola, and then graphing as usual.

Question 2

• The general equation of a conic with axes not parallel to the xy-axes is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.

Answer 2

• The general equation of a conic with axes not parallel to the xy-axes is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.

Question 3

• If the equation of a conic includes a Bxy-term, then that conic’s axes are not parallel to the xy-axes.

Answer 3

• If the equation of a conic includes a Bxy-term, then that conic’s axes are not parallel to the xy-axes.

Question 4

• Steps for Graphing a Conic when the Equation Contains a Bxy-term
1. Find α, the angle of rotation needed to make the xy-axes parallel to the conic’s axes.
2. Use the formulas for the coefficients and α to write the equation of the conic in terms of x’ and y’,
A’(x’)2 + C’(y’)2 + D’x’ + E’y’ + F’ = 0.
3. Graph the resulting equation and then rotate the axes by the angle α.

Answer 4

• Steps for Graphing a Conic when the Equation Contains a Bxy-term
1. Find α, the angle of rotation needed to make the xy-axes parallel to the conic’s axes.
2. Use the formulas for the coefficients and α to write the equation of the conic in terms of x’ and y’,
A’(x’)2 + C’(y’)2 + D’x’ + E’y’ + F’ = 0.
3. Graph the resulting equation and then rotate the axes by the angle α.