Unit 4 Curve Tracing
Introduction
For evaluating areas, volumes of revolution etc. we need to know the general
nature of the given curve. Given the equation of a curve explicitly as y = f ( x ) or
implicitly as g ( x, y ) = c, a constant, many properties of the curve can be determined
easily by knowing its graph. In this chapter we shall study the methods of tracing a curve
in general whose equation is given in Cartesian or in polar form and the properties of
some standard curves commonly met in engineering problems.
Common Curves
You have already studied quite in detail straight line, circle, parabola, ellipse and
hyperbola, including rectangular hyperbola. We shall briefly review these curves.
a) Straight Line: General equation of straight line is of the form ax + by + c = 0.
To plot a straight line we put y = 0 and find x. Also we put x = 0 and find y. We plot
these two points and join them to get the required line. Some particular cases of straight
Lines are shown below.
b) Circle: General equation of circle is x 2 + y2 + 2gx + 2fy + c = 0. Its centre is ( -g, -f )
and radius = g 2 f 2 c . Circle with centre at origin and radius a
and Circle with centre at ( -g, -f ) and radius g 2 f 2 c is shown below.
c) Parabola : General equation of parabola is y = ax2 + bx + c or x = ay2 + by + c.
d
x
x= a
,By completing the square on x or on y and by shifting the origin the equation can be
written in standard form as y2 = 4ax or x2 = 4by.
x2 y2
d) Ellipse : The equation of ellipse in standard form is 2 2 1
a b
e) Hyperbola : Equation of hyperbola in standard form is
Tangents at the origin and at other points
If more than one branch of the curve passes through a point, then that point is
called a multiple point of the curve. A double point is a point through which two
branches of the curve passes. A double point is called a cusp if two branches of the
curve have the same tangents. A double point is called a node if two branches of the
curve have distinct tangents. An isolated point of a curve is a point such that there
exists a neighborhood of that point in which no other point of the neighborhood lies in
the region of existence.
Asymptotes
A straight line is said to be an asymptote to an infinite branch of a curve if the
perpendicular distance from a point on the curve to the given line approaches to zero as
the point moves to infinity along the branch of curve.
The asymptotes parallel to x - axis are called horizontal asymptotes, those which
are parallel to y - axis are called vertical asymptotes and those which are neither parallel
to x - axis nor parallel to y - axis are called oblique asymptotes.
, The equations of horizontal asymptotes are obtained by equating the coefficient of
highest degree term in x to zero if it is not a constant. The equations of vertical asymptotes
are obtained by equating the coefficient of highest degree term in y to zero if it is not a
constant. To obtain the equations of oblique asymptotes, substitutes y = mx + c in the given
equation. Then equate the coefficients of the highest degree term in x and next highest
degree term in x to zero, if it is not a constant, to determine m and c. If the values of m and
c exists, then y = mx + c is the equation of the oblique asymptote.
Derivatives
For the equation of the curve, determine dy/dx.
• If dy/dx > 0 in an interval then the curve increases in that interval.
• If dy/dx < 0, then the curve decreases.
• If dy/dx = 0, at a point, then the curve has a stationary point.
• If dy/dx = 0, at a point, then tangent at that point is parallel to x - axis.
• If dy/dx = , at a point, then tangent at that point is parallel to y - axis.
0
• If dy/dx = , at a point, then apply L - Hospital’s Rule.
0
Procedure for tracing curves given· in Cartesian equations
I) Symmetry: Find out whether the curve is symmetrical about any line with the help of
the following rules:
a) The curve is symmetrical about the x - axis if the equation of the curve
remains unchanged when y is replaced by - y i.e. if the equation contains only
even powers of y.
b) The curve is symmetrical about the y - axis if the equation of the curve
remains unchanged when x is replaced by - x i.e. if the equation contains only
even powers of x.
c) The curve is symmetrical in opposite quadrants if the equation of the curve
remains unchanged when both x and y are replaced by - x and - y.
d) The curve is symmetrical about the line y = x if the equation of the curve
remains unchanged when x and y are interchanged.
e) The curve is symmetrical about the line y = - x if the equation of the curve
remains unchanged when both x and y are replaced by - y and - x.
II) Origin : Find out whether the origin lies on the curve. If it does, find the equations
of the tangents at the origin by equating to zero the lowest degree terms.
Introduction
For evaluating areas, volumes of revolution etc. we need to know the general
nature of the given curve. Given the equation of a curve explicitly as y = f ( x ) or
implicitly as g ( x, y ) = c, a constant, many properties of the curve can be determined
easily by knowing its graph. In this chapter we shall study the methods of tracing a curve
in general whose equation is given in Cartesian or in polar form and the properties of
some standard curves commonly met in engineering problems.
Common Curves
You have already studied quite in detail straight line, circle, parabola, ellipse and
hyperbola, including rectangular hyperbola. We shall briefly review these curves.
a) Straight Line: General equation of straight line is of the form ax + by + c = 0.
To plot a straight line we put y = 0 and find x. Also we put x = 0 and find y. We plot
these two points and join them to get the required line. Some particular cases of straight
Lines are shown below.
b) Circle: General equation of circle is x 2 + y2 + 2gx + 2fy + c = 0. Its centre is ( -g, -f )
and radius = g 2 f 2 c . Circle with centre at origin and radius a
and Circle with centre at ( -g, -f ) and radius g 2 f 2 c is shown below.
c) Parabola : General equation of parabola is y = ax2 + bx + c or x = ay2 + by + c.
d
x
x= a
,By completing the square on x or on y and by shifting the origin the equation can be
written in standard form as y2 = 4ax or x2 = 4by.
x2 y2
d) Ellipse : The equation of ellipse in standard form is 2 2 1
a b
e) Hyperbola : Equation of hyperbola in standard form is
Tangents at the origin and at other points
If more than one branch of the curve passes through a point, then that point is
called a multiple point of the curve. A double point is a point through which two
branches of the curve passes. A double point is called a cusp if two branches of the
curve have the same tangents. A double point is called a node if two branches of the
curve have distinct tangents. An isolated point of a curve is a point such that there
exists a neighborhood of that point in which no other point of the neighborhood lies in
the region of existence.
Asymptotes
A straight line is said to be an asymptote to an infinite branch of a curve if the
perpendicular distance from a point on the curve to the given line approaches to zero as
the point moves to infinity along the branch of curve.
The asymptotes parallel to x - axis are called horizontal asymptotes, those which
are parallel to y - axis are called vertical asymptotes and those which are neither parallel
to x - axis nor parallel to y - axis are called oblique asymptotes.
, The equations of horizontal asymptotes are obtained by equating the coefficient of
highest degree term in x to zero if it is not a constant. The equations of vertical asymptotes
are obtained by equating the coefficient of highest degree term in y to zero if it is not a
constant. To obtain the equations of oblique asymptotes, substitutes y = mx + c in the given
equation. Then equate the coefficients of the highest degree term in x and next highest
degree term in x to zero, if it is not a constant, to determine m and c. If the values of m and
c exists, then y = mx + c is the equation of the oblique asymptote.
Derivatives
For the equation of the curve, determine dy/dx.
• If dy/dx > 0 in an interval then the curve increases in that interval.
• If dy/dx < 0, then the curve decreases.
• If dy/dx = 0, at a point, then the curve has a stationary point.
• If dy/dx = 0, at a point, then tangent at that point is parallel to x - axis.
• If dy/dx = , at a point, then tangent at that point is parallel to y - axis.
0
• If dy/dx = , at a point, then apply L - Hospital’s Rule.
0
Procedure for tracing curves given· in Cartesian equations
I) Symmetry: Find out whether the curve is symmetrical about any line with the help of
the following rules:
a) The curve is symmetrical about the x - axis if the equation of the curve
remains unchanged when y is replaced by - y i.e. if the equation contains only
even powers of y.
b) The curve is symmetrical about the y - axis if the equation of the curve
remains unchanged when x is replaced by - x i.e. if the equation contains only
even powers of x.
c) The curve is symmetrical in opposite quadrants if the equation of the curve
remains unchanged when both x and y are replaced by - x and - y.
d) The curve is symmetrical about the line y = x if the equation of the curve
remains unchanged when x and y are interchanged.
e) The curve is symmetrical about the line y = - x if the equation of the curve
remains unchanged when both x and y are replaced by - y and - x.
II) Origin : Find out whether the origin lies on the curve. If it does, find the equations
of the tangents at the origin by equating to zero the lowest degree terms.
