Conic sections are fascinating shapes formed by slicing a cone. They include circles, ellipses, parabolas, and hyperbolas. Understanding their equations and properties is key to analyzing their behavior and applications in various fields.
Rotating axes can simplify conic equations by eliminating the xy term. This transformation helps reveal the true nature of the conic, making it easier to identify its type and characteristics. It's a powerful tool for simplifying complex geometric problems.
Conic Sections and Rotation of Axes
General form of conic sections
General form equation of a : Ax2+Bxy+Cy2+Dx+Ey+F=0
B=0 and A=C: circle (e.g., x2+y2=25)
B=0 and A=C: ellipse (e.g., 4x2+y2=16)
B=0 and either A=0 or C=0: parabola (e.g., y2=4x)
B=0 and B2−4AC=0: parabola (e.g., xy=1)
B=0 and B2−4AC<0: ellipse (e.g., x2+4xy+y2=1)
B=0 and B2−4AC>0: hyperbola (e.g., x2−4xy+y2=1)
Rotation of axes for conics
Eliminates the xy term in the general form equation
θ given by tan2θ=A−CB
New coordinates (x′,y′) related to original coordinates (x,y):
x=x′cosθ−y′sinθ
y=x′sinθ+y′cosθ
Substitute expressions into general form equation and simplify to obtain equation in new coordinate system (e.g., x2+4xy+y2=1 becomes 2x′2+y′2=1)
This process is an example of an orthogonal transformation
Standard form of rotated conics
After rotation of axes, resulting equation in form A′x′2+C′y′2+D′x′+E′y′+F′=0
Convert to standard form by completing the square for x′ and y′ terms:
For x′ term, add and subtract (2A′D′)2 to equation
For y′ term, add and subtract (2C′E′)2 to equation
Resulting equation in standard form of conic section:
Parabola: (y′−k′)2=4p′(x′−h′) or (x′−h′)2=4p′(y′−k′) (e.g., (y′−2)2=8(x′−1))
Hyperbola: a′2(x′−h′)2−b′2(y′−k′)2=1 or a′2(y′−k′)2−b′2(x′−h′)2=1 (e.g., 9(x′−2)2−4(y′+1)2=1)
Analysis of non-rotated conics
Use original general form equation Ax2+Bxy+Cy2+Dx+Ey+F=0
Find center of conic by solving system of equations:
2Ax+By+D=0
Bx+2Cy+E=0
Determine type of conic using discriminant B2−4AC and values of A and C (as mentioned in general form section)
Analyze conic's properties (foci, vertices, asymptotes) using original equation and center coordinates (e.g., for 4x2+y2=16, center is (0,0), vertices are (±2,0), and foci are (±3,0))
Advanced Topics in Rotation of Axes
Linear algebra provides a powerful framework for understanding rotation of axes
Rotation can be represented using matrix multiplication
Eigenvalues and eigenvectors play a crucial role in determining the principal axes of conics
The principal axis theorem relates to finding the simplest form of quadratic forms through rotation
Key Terms to Review (4)
Angle of rotation: The angle of rotation is the measure of the angle by which a figure is rotated about a fixed point, often the origin. It is typically measured in degrees or radians.
Conic section: A conic section is a curve obtained by intersecting a cone with a plane. The four types are circles, ellipses, parabolas, and hyperbolas.
Degenerate conic sections: Degenerate conic sections are special cases of conic sections that result when the plane intersects the cone in a manner that produces simpler geometric figures like points, lines, or pairs of lines. They occur when the discriminant of a conic section's equation is zero.
Nondegenerate conic sections: Nondegenerate conic sections are the curves obtained from the intersection of a plane and a double-napped cone, which include ellipses, parabolas, and hyperbolas. They are distinct from degenerate conics such as points, lines, or intersecting lines.