CHAPTER 10 CONICS AND POLAR COORDINATES

10.1 The Parabola In a plane with line, l, (directrix) and fixed point F (focus), eccentricity is defined as the ratio of the distance from any point, P, to the focus to the distance to the directrix.

Parbola, e=1 Set of point P such that the distance from a point to the focus = distance from point to the directirx. Standard equation of a parabola

10.2 Ellipses & Hyperbolas

Standard Equation

10.3 Translation & Rotation of Axes Conics need not be centered at the origin. They could be centered at any point: (h,k) Let u= x – h, v = y – k Equivalently: x = u + h, y = v + k May need to complete the square to create standard form for recognition of conic.

Rotation of Axes The xy-axes may be rotated through angle theta for any conic How is the angle, theta, found?

10.4 Parametric Representation of Curves in the Plane For a parametric function, x=f(t), y=g(t) Values of t as t advances from a to b, define where the curve begins and ends

Differentiation of parametric equations Let f & g be continuously differentiable with f’(t) not equal 0 on a<t<b. Then x=f(t) and y=g(t) The derivative of y with respect to x is:

Calculation of arc length

10.5 Polar Coordinate System Given a fixed point (O), the pole or origin, a polar axis running horizontally to the right of the origin, any point can be defined as: distance, r, from the origin, rotated through an angle, theta, from the polar axis. The coordinates of the point are of the form: (r, theta)

Relationships between polar & cartesian coordinates Polar to Cartesian Cartesian to Polar

Example: Show that the given polar equation is that of an ellipse:

Polar form of conics If a conic has its focus at the pole and its directrix d units away, the final form is:

10.6 Graphs of Polar Equations Common polar graphs: –Cardiods –Limacons –Lemniscates –Roses –Spirals

Limacons & Cardiods If a=b, cardiod (heart-shaped) If a<b, inner loop

Lemniscates Figure-8 shaped curves

Roses Polar equations of the form: n leaves (n odd) 2n leave (n even)

Spiral of Archimedes and Logarithmic Spiral

10.7 Calculus in Polar Coordinates Area in polar coordinates Tangents in polar coordinates

Area in polar coordinates Recall how to find area of sector of a circle: For a polar curve, r = f(theta) and the angle changes as you move along the curve from a to b. The area is the sum of all the areas of each little sector, which is an integral:

Tangents in Polar Coordinates In Cartesian coordinates, m = dy/dx