MTH301 Lectures 23-45 by Orange Monkey

MTH301 FINAL TERM NOTES · LECTURE 23 TO 45 PROVIDE BY · ORANGE MONKEY TEAM · Lecture No – 23 Polar Coordinate Systems

POLAR COORDIANTE SYSTEMS To form a polar coordinate system in a plane, we pick a fixed point O, called the origin or pole, and using the origin as an endpoint we construct a ray, called the polar axis. WHAT IS POLAER ANGLE? The number r is called the radial distance of P and θ is called a polar angle of P

(5, 120°), and (4, 330°) are plotted in polar coordinate systems. Using formulas x = r cos θ y = r sin θ x2 +y2 = r2 y/ x =tanθ LINES IN POLAR COORDIANTES A line perpendicular to the x-axis and passing through the point t with ‘XY’ coordinates with (a,0) has the equation x=a . We obtain the general polar form of the line. r (Acosθ + B sinθ) + C = 0 CIRCLES IN POLAR COORDINATES

Lecture No -24 Sketching

Sketching of Curves in Polar Coordinates

•  Symmetry about the Initial Line If the equation of a curve remains unchanged when (r, θ) is replaced by

either (r,-θ ) in its equation ,then the curve is symmetric about initial line.

•  Symmetry about the y-axis If when (r, θ) is replaced by either (r,π −θ ) in The equation of a curve and an equivalent equation is obtained ,then the curve is symmetric about the line perpendicular to the initial i.e, the y-axis.
•  Symmetry about the Pole If the equation of a curve remains unchanged when either (-r, θ) or is substituted for (r, θ) in its equation ,then the curve is symmetric about the pole. In such a case, the center of the curve. (R, θ)

(R, -θ) Position of the Pole Relative to the Curve See whether the pole on the curve by putting r=0 for in the equation of the curve and solving θ. Table of Values

•  Table of Values Construct a sufficiently complete table of values. This can be of great help in sketching the graph of a curve.
•  When r = 0 , θ = 0 .Hence the curve passes through the pole.

Difference b/w CARDIDOIDS AND LIMACONS The equations of above form produce polar curves called limacons. Because of the heart shaped appearance of the curve in the case a = b, limacons of this type are called cardioids. r=a+b sin θ, r = a−bsin θ r =a+b cos θ, r = a−bcos θ LEMINSCATE If a > 0, then Equation of the form r 2 = a2 cos2θ, r2 = − a2 cos 2θ r2 = a2 sin 2θ, r2 = − a2 sin 2θ Represent propeller-shaped curves, called lemiscates