Curve Tracing
Multiple point : If number of branches of the curve passes
through a point, then that point is called multiple point.
Through multiple point, more than one branches of the curve
passes. If n branches of the curve pass through a point, then
such point is called a multiple point of nth order.
Double point : A point, through which only two branches of the curve passes, is called
double point.
Node : A double point P is called node, if the branches of the curve
pass, through it are real and the tangents at the point of intersection
are real and different.
Cusp : A double point P is called cusp if the tangents at the point of intersections are real
and coincides to each other.
Conjugate point : The point P is called conjugate or isolated point if there are no any real
point in its neighborhood. Conjugate point satisfies the equation of curve but it does not
lies on curve.
Singular points : The point of inflexion, node, cusp, isolated point,
multiple point etc. are called singular points.
CARTESIAN CURVES : Cartesian curves are in the form of
Explicit curves: (i) y = f (x) (ii) x = f (y)
Implicit Curves: f (x, y) = 0
Rules to Trace Cartesian Curves:
(I) Symmetry : (Symmetry means mirror image)
Symmetric about X-axis : In the given equation, if all powers of y are even then the
curve is symmetric about X-axis.
Symmetric about Y-axis :In the given equation, if all powers of x are even then the
curve is symmetric about Y-axis.
Symmetric about y = x line : By interchanging x and y i.e. put x = y and y = x, if the
given equation is remain unchanged then the given curve is symmetric about y = x
line.
Symmetric about y = - x line : By interchanging x and - y i.e. put x = - y and y = - x, if
the given equation is remain unchanged then the given curve is symmetric about y = - x
line.
Symmetric in opposite quadrants (or at origin) : By interchanging x by – x and y by
– y, if the given equation remain unchanged, then the curve is symmetrical in opposite
quadrants.
(II) Points of Intersection and Origin :
• With x-axis : Put y = 0 in the given equation and find the values of x;
we get (a1, 0), (b1, 0)… are the points of intersection with X-axis.
• With Y-axis : Put x = 0 in the given equation and find the values of y,
we get (0, a2), (0, b2) … are the points of intersection with Y-axis.
• Origin : Put x = 0 in the given equation if we get, y = 0 then the curve passes through
origin.
• Tangent at origin : If the curve passes through origin, then find
tangent at origin by using Newton’s method.
“By equating the lowest degree term (or terms) with zero”
we get equation of tangent at origin.
(III) Region of absence : (Nature of Curve) :
(i) If the given curve is symmetric about X – axis then express the equation in the explicit
form (say) y = f (x) and check how y varies as x varies.
If y2 < 0 (y is imaginary) for some value of x > a (say)
then the curve does not exists for x > a.
(ii) If the given curve is symmetric about Y – axis then express the given equation as
x = f (y) and check how x varies as y varies continuously.
If x2 < 0 (x is imaginary) for some value of y > b (say)
then the curve does not exists for y > b.
(IV) Asymptote : Asymptote means the tangent to the curve at infinity.
A curve of degree ‘n’ have maximum ‘n’ asymptotes.
(i) Asymptote parallel to X-axis : By equating, the co-efficient of highest power of x to
zero, we get the equation of asymptote parallel to X-axis.
(ii) Asymptote parallel to Y-axis : By equating the co-efficient of highest power of y to
zero, we get equation of asymptote parallel to Y-axis.
(iii) Oblique asymptote : (Oblique asymptote means asymptote neither parallel to X-axis
nor parallel to Y-axis).
To find Oblique asymptote Put , y = mx + c in the given equation and then equate the
coefficients of the two highest powers of x to zero, we get the values of m and c.
(V) Tangents: (Special point) : By using the following rules, we can find the tangent other
than origin point.
𝑑𝑦
1.If = 0, at some point (a, b), then tangent at point (a, b) is parallel to X-axis.
𝑑𝑥
𝑑𝑦
2. If = ∞, at some point (a, b), then tangent at point (a, b) is parallel to Y-axis.
𝑑𝑥
𝑑𝑦
3.If > 0, in the interval (a, b) then the curve strictly increasing in the interval (a, b)
𝑑𝑥
𝑑𝑦
4. If < 0, in the interval (a, b) then the curve strictly decreasing in the interval (a, b)
𝑑𝑥
5. The curve attains its maximum or minimum values at the points
𝑑𝑦
where = 0.
𝑑𝑥
𝑑𝑦
Note: Always find at points of intersections with co-ordinate axes &
𝑑𝑥
line of symmetry (if any)
Polar Curves:
In general, the curves in polar form can be written as r = f ().
In polar system,
(i) Pole : In the Polar curves the fixed point is called pole
(Here, origin O is a pole).
(ii) Initial line : A fixed line (positive x-axis) is called initial line.
(iii) Radius vector : The distance of a point from pole is called radius vector. ( Here, OP = r ).
(iv) Vectorial angle () : The angle made by radius vector with initial line is called
vectorial angle. ( XOP = ) And is the angle made by tangent with radius vector (r).
(v) = /2, represent a straight line passes through pole and
perpendicular to initial line (i.e. Y-axis)
(vi) If it is difficult to trace the curve in Cartesian form, we may convert it into polar form
by using the polar transformations as :
𝑦
x = r cos ; y = r sin and r2 = x2 + y2 and θ = 𝑡𝑎𝑛 −1
𝑥
and then trace the curve by using the rules in polar form.
Rules to Trace Polar Curves:
(I) Symmetry :
(i) About initial line : By replacing by – , if the given equation remains unchanged,
then the curve is symmetric about initial line (x-axis)
(ii) About pole : If the powers of r are even, then the curve is symmetrical about pole.
(Also, by putting r by – r, if given equation remains same)
𝝅
(iii) About = line : By replacing = – and r = – r at the same time, if the given
𝟐
𝝅
equation remains unchanged, then the curve is symmetrical about the line = (Y axis )
𝟐
𝝅
( = is the line through pole perpendicular to the initial line)
𝟐
𝝅
(iv) About = line : If the equation remains unchanged by replacing by – , then
𝟐
𝝅
also the curve is symmetrical about the line = .
𝟐
(v) About pole (or opposite quadrants) :
If the given equation remains unchanged by putting r = – r or = + then the curve is
symmetric in opposite quadrant (or pole).
𝝅 𝜋
(vi) About the line 𝜽 = : If the given equation remains unchanged by putting = – ,
𝟒 2
𝝅
then the curve is symmetric about the line 𝜽 = .
𝟒
II. Pole : Put, r = 0 in the equation of polar curve we get a equation of tangent
to the curve at pole.
III. Limits: If possible find the greatest and lowest values of r.
𝑑𝜃
IV. Tangents: If 𝜙 is the angle between radius vector & tangent then tan 𝜙 = 𝑟
𝑑𝑟
V. Table: Make a table for different values for 𝜃, 𝑟 𝑎𝑛𝑑 tan 𝜙
𝜽 𝝅 𝝅 𝟑𝝅 𝝅 𝟓𝝅 𝟑𝝅 𝟕𝝅 𝟐𝝅
𝟒 𝟐 𝟒 𝟒 𝟐 𝟒
r
tan 𝜙
Note: Rose Curves 𝑟 = acos 𝑛𝜃 𝑎𝑛𝑑 𝑟 = 𝑎 sin 𝑛𝜃
• If n is even then there are 2n similar loops in the curve
• If n is odd then there are n similar loops in the curve
𝜽 𝟎 𝝅 𝟐𝝅 𝟑𝝅 𝟒𝝅 𝟓𝝅 …up to 𝟐𝝅 or depend upon Symmetry
𝟐𝐧 𝟐𝐧 𝟐𝐧 𝟐𝐧 𝟐𝐧
r
tan 𝜙
Rules to Trace Parametric Curves:
The parametric curve equation as the form 𝑥 = 𝑓1 𝑡 , 𝑦 = 𝑓2 𝑡 where t is a parameter
I. Symmetry:
If 𝑥 = 𝑓1 𝑡 is even & 𝑦 = 𝑓2 𝑡 is odd, then curve is symmetric about x-axis.
If 𝑦 = 𝑓2 𝑡 is even & 𝑥 = 𝑓1 𝑡 is odd, then curve is symmetric about y-axis.
If 𝑥 = 𝑓1 𝑡 𝑎𝑛𝑑 𝑦 = 𝑓2 𝑡 both are odd, then curve is symmetric in opposite quadrants.
II. Origin: If for 𝑡 = 0, we get 𝑥 = 0 𝑎𝑛𝑑 𝑦 = 0 , then curve pass through origin.
III. Limits: Find greatest & lowest values of y & x (if possible).
𝑑𝑦
𝑑𝑦
IV. Tangents: Find = 𝑑𝑥 𝑑𝑡
𝑑𝑥 𝑑𝑡
𝑑𝑦
If = 0 for some value of ‘t’ then tangent llel to x-axis at that point.
𝑑𝑥
𝑑𝑦
If = ∞ for some value of t then tangent llel to y-axis at that point.
𝑑𝑥
𝑑𝑦
V. Table: Make a table for different values of t, x, y,
𝑑𝑥
𝐭 0 𝝅 𝝅 𝟑𝝅 𝟐𝝅
𝟐 𝟐
x
𝑦
𝑑𝑦
𝑑𝑥
Note: Cycloid Curves:
1. 𝑥 = 𝑎 𝜃 + sin 𝜃 ; 𝑦 = 𝑎 1 + cos 𝜃
2. 𝑥 = 𝑎 𝜃 − sin 𝜃 ; 𝑦 = 𝑎 1 − cos 𝜃
3. 𝑥 = 𝑎 𝜃 + sin 𝜃 ; 𝑦 = 𝑎 1 − cos 𝜃
4. 𝑥 = 𝑎 𝜃 − sin 𝜃 ; 𝑦 = 𝑎 1 + cos 𝜃
