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conics

Conics

Subject: 
Trigonometry
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geometry
Algebraic geometry
Conic sections
Analytic geometry
Curves
Birational geometry
Parabola
Hyperbola
Ellipse
focus
Pole and polar
Euclidean geometry
conics

Analytic geometry is roughly the same as plane geometry except that in analytic geometry, figures are studied in the context of the coordinate plane. Instead of focusing on the congruence of shapes like plane geometry, analytic geometry deals with the coordinates of shapes and formulas for their graphs in the coordinate plane. Much of analytic geometry focuses on the conics. A conic is a two-dimensional figure created by the intersection of a plane and a right circular cone. All conics can be written in terms of the following equation: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 . The four conics we'll explore in this text are parabolas, ellipses, circles, and hyperbolas.

Trig

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Trigonometry
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conics
rotated axis

10.5 Notes - Rotation of Axes At times we may be faced with an equation of a conic section whose axes would not be parallel to the x or y-axes. (These conics have an equation of the form where .) As a result it may be difficult to ascertain the nature of the graph or its position on the coordinate system. In this case we may simplify the equation by a process of rotating the axes. ? In the figure the axes have been rotated through an acute angle about the origin to produce a new pair of axes, which we will call the X and Y axes. A point P that has coordinates (x, y) in the old system has coordinates (X,Y) in the new system. Let r equal the distance of P from the origin. Let be the angle that segment OP makes with the X axis. ? ?

TRIG rotated axis proof

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Trigonometry
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conics
rotated axis

10.5B Notes ? Rotation of Axes The general equation of the second degree in two variables may be written where A, B, C, D, E and F are real coefficients and not all of A, B or C are zero. We can transform this equation into an equation in terms of X and Y by rotating the axes through an appropriate angle ?. To find the angle that works substitute for x and y using the rotation formulas. x = X cos ? ? Y sin ?, y = X sin ? + Y cos ? in terms of X and Y is Expanding this and collecting like terms, (which is quite a job), we obtain an equation of the form ?.. Where In order to eliminate the XY-term, we need to choose so that . That is or equivalently . Note: Don?t forget your half angle formulas for sine and cosine. ?
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