Course Outline
& Rules for tracing Cartesian and polar curves.
& Tracing curves of some well known curves.
Introduction
To trace the curve whose equation is given in Cartesian form ;
1. Symmetry
– If the
equation of curve contains only the even power of x, then the curve is
symmetrical about y- axis. e.g. x2 = 4ay.
– If the
equation of the curve contains only the even power of y, then the curve is
symmetrical about x- axis. e.g. y2 = 4ax.
– If the
equation of curve remains unchanged when x and y are interchanged, then the
curve is symmetrical about the line y = x. e.g. x3 + y3 = 3axy.
– If the
curve remains unchanged when x is replaced by –y and y is replaced by –x, then
the curve is symmetrical about opposite quadrants or the line y = –x. e.g. xy =
4.
2. Origin
If
the point (0, 0) satisfies the equation then the curve passes through the
origin. e.g. x2 = 4ay, x3 + y3 = 3axy etc.
3. Noticeable points
– Putting
x = 0, the value of y gives the point where curve cuts the
y- axis.
– Putting
y = 0, the value of x gives the point where the curve meets
x- axis.
4. Tangents
– If the
curve passes through the origin, then the tangent at the origin are obtained by
equating to zero the lowest degree term in x and y.
– The
tangent at (h, k) are obtained by shifting the origin at (h, k) and equating to
zero the lowest degree term.
– The
equation of tangents at any point may be obtained by finding the value of dy/dx
at that point.
5. Region
To find the region, for any value of x the values y must be real and for
any value of y the value of x must be real for example y2 = 4ax, the curve lies entirely on the right hand side of y-
axis and y = x3, the curve extends from
–¥ to ¥.
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