1. CURVE TRACING

2. Different forms of CURVES
CARTESIAN FORM:
POLAR FORM:
PARAMETRIC FORM:
PEDAL FORM:
INTRINSIC FORM:
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3. CARTESIAN CURVES
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4. SOAP
Symmetry:
About X axis : Every where in the equation powers of y are even.
About Y axis : Every where in the equation powers of x are even.
In opposite quadrants or with respect to origin : On replacing x by –x and y by -y in the equation of the curve, no change in the equation.
About the line y = x : Replace x by y and y by x no change in the equation.
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5. SOAP
Origin:
If the independent constant term is absent in the equation of the curve then it passes through the origin. i.e. if x=0 and y=0 satisfies the equation of the curve then it passes through the origin.
If the curve passes through the origin, then the Tangents to the curve at origin can be obtain by equating to zero lowest degree term of the equation.
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6. SOAP
Asymptote:
A tangent to the curve at infinity is called its asymptote.
Asymptote can be classified into two category
Parallel to X-axis or parallel to Y-axis.
Oblique Asymptote: Neither Parallel to X-axis nor parallel to Y-axis.
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7. 1. Asymptote parallel to Y-axis
Asymptote Parallel to Y-axis can be obtained by equating to zero the coefficient of highest degree term in y.
2. Asymptote parallel to X-axis
Asymptote Parallel to X-axis can be obtained by equating to zero the coefficient of highest degree term in x.
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8. Procedure: Let y = mx + c be the asymptote.
3. Oblique Asymptote:
If the Equation of the curve is in the implicit form i.e. f(x, y) = 0, then it may have oblique asymptote.
Procedure: Let y = mx + c be the asymptote.
Solving f(x, mx+c) = 0 and equating the co-efficients of highest and second highest power of x to 0 we get required values of m and c.
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9. SOAP POINTS Find X-Intercept Or Y-Intercept:
X-Intercept can be obtained by putting y = 0 in the equation of curve.
Y-Intercept can be obtained by putting x = 0 in the equation of curve.
For the Curves ,
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10. If =0, then tangent will be parallel to X-axis.
If = , then tangent will be parallel to Y-axis.
If > 0, in the Interval a < x < b, then y is increasing function in a < x < b.
If < 0, in the Interval a < x < b, then y is decreasing function in a < x < b.
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11. Region of Absence:
If y2<0, for x < a (x=a is an X-intercept), then there is no curve to the left of the line x=a.
If y2<0, for a < x < b (a and b are X-intercepts) then there is no curve between the lines x = a and x = b.
If y2<0, for x > b ( x = b is an X-intercept) then there is no curve to the right of the line x=b.
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19. POLAR CURVES
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20. Symmetry::
About the initial line: Replace by no change in the equation.
About Pole: Replace r by -r no change in the equation.
About the line perpendicular to the initial line (Y axis): Replace by and r by - r. OR replace by no change in the equation.
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21. Prepare table of values of r and .
Pole :: If for certain value of , r=0 the curve will pass through the pole.
Tangents at pole: Equate r=0 and solve for , lines =constant are tangents at the pole.
Prepare table of values of r and .
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22. tangent and radius vector coincides and
Angle between tangent and radius vector:: be angle between tangent and radius vector then ,
tangent and radius vector coincides and
tangent and radius vector are perpendicular.
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23. Region of absence:
If for  <  <, r2 <0, then no curve lie between  = , = .
|sin |  1, |cos  | 1 .So for the curves r = acos n , r = asin n ,r a. So, no curve lies outside the circle of radius r.
In most of the polar equations, only functions sin , cos  occurs and so values of  between 0 to 2 should be considered. The remaining values of , gives no new branch of the curve.
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24. Polar Curves
Type I
Sine curve can be obtained from corresponding cosine by rotating the plane through an angle /2n.
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26. n = – ½ Parabola
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28. n = 2 Bernoulli’s Lemniscates
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29. n = 1/2 Cardiode
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30. Type II Rose Curves Type III Type IV Spirals 1/18/2019
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36. PARAMETRIC CURVES
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37. Symmetry ::
About X axis – Replace t by –t , x remain unchanged and y changes the sign then curve is symmetric about X- axis.
About Y axis - Replace t by –t , x changes the sign and y remain unchanged then curve is symmetric about Y- axis.
Note that :: For trigonometric equation replacing t by , y remains unchanged and x changes the sign curve will be symmetric about Y- axis.
About Origin - Replace t by –t both x and y changes the sign curve is symmetric about origin.
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38. Points of Intersections ::
If for some value of t, both x and y are zero curve passes through origin.
Find X and Y intercepts.
Find asymptotes if any.
Find region of absence.
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39. Tangents
Find
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45. WITCHES OF AGNESI
The curve is obtained by drawing a line OB from the origin through the circle of radius a and center (0,a), then picking the point with the y coordinate of the intersection with the circle and the x coordinate of the intersection of the extension of line OB with the line y=2a .
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46. CYCLOID
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47. EPICYCLOID
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48. CARDIOID
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49. PASCAL’S LIMACON
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50. PASCAL’S LIMACON
Let P be a point and C be a circle whose center is not P. Then the envelope of those circles whose center lies on C and that pass through P is a limaçon.
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51. HYPOCYCLOID
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52. ASTROID
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55. THANK YOU
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