1. Symmetry
(i) A curve is symmetrical about xaxis if the equation remains the same by replacing y by
–y. here y should have even powers only.
For example y^{2} = 4ax^{.}
(ii) It is symmetrical about yaxis if it contains only even powers of x For example x^{2} = 4ay.
(iii)If on interchanging x and y, the equation remains the same then the curve is
symmetrical about the line. y = x, For example x^{3 + y3 = 3axy }
(iv) A curve is symmetrical in the opposite quadrants if its equation remains the
same when x and y replaced by –x and –y. For example y = x^{3
}
2. (a) Curve through Origin
The curve passes through the origin, if the equation does not contain constant term.
For example the curve y^{2 }= 4ax passes through the origin.
2 (b)Tangents at the origin:
To know the nature of a multiple point it necessary to find the tangent at that point.
The equation of the tangent at the origin can be obtained by equating to zero, the lowest degree term in the equation of the curve.
3. The points of intersection with the axes
(i) By putting y= 0 in the equation of the curve we get the coordinates of the point of intersection with the x –axis.
(ii) By putting x = 0 in the equation of the curve; the ordinate of the point of intersection with the yaxis is obtained by solving the new equation.
4. Regions in which the curve does not lie.
If the value of y is imaginary for certain value of x then the curve does not exist for such values.
Example. y^{2} = 4x
Example. a^{2}x^{2} = y^{3}(2a  y).
For negative value of x, y is imaginary so there is no curve is second and third quadrant.
 (i) For y> 2a x is imaginary so there is no curve in second and third quadrant
 (ii) For negative values, of y, x is imaginary. There is no curve in 3rd and 4th quadrant.
5. Asymptotes are the tangents to the curve at infinity.
(a) Asymptote parallel to the xaxis is obtained by equating to zero, the coefficient of the highest power of x.
For example yx^{2}  4x^{2 }+ x + 2 = 0^{ }
(y  4)x^{2} + x + 2 = 0
The coefficient of the highest power of x i.e. x2 is y  4 = 0
y  4 = 0 is the asymptote parallel to the x axis.
(b) Asymptote parallel to the yaxis is obtained to zero, the coefficient of highest power of y.
For example
xy3  2y3 + y2 + x2 + 2 = 0
(x2)y3 + y2 + x2 + 2 = 0
The coefficient of the highest power of y. i.e. y3 is x 2.
X 2 = 0 is the asymptote parallel to yaxis.
6. Tangent.
Put 0=dx/dy for the points where tangent is parallel to the xaxis.
For example x^{2 }+ y^{2 } 4x + 4y 1 = 0^{ ………(1)
} 2x +2ydxdy 4 + 4dxdy= 0
^{ }(2y+4) dx/dy= 4  2x or dx/dy=4224+−yx
^{ }Now dxdy= 0. 42x = 0 or x = 2
Putting x =2 in (1), we get y2 + 4y 5 = 0
y= 1, 5
The tangents are parallel to xaxis at the points (2,1) and (2, 5).
7. Table.
Prepare a table for certain values of x and y and draw the curve passing through them.
For example y^{2} = 4x +4
x

1

0

1

2

3

Y

0

±2

±22

±23

±4

Example. Trace the curve y^{2}( 2a – x ) = x^{3}
Solution: y^{2}= x3/( 2a – x ) ……..(1)
(i) Origin: Equation does not contain any constant term; therefore, it passes through origin.
(ii) Symmetrical about xaxis: Equation contains only even powers of y; therefore, it is symmetrical about xaxis.
(iii)Tangent at the origin: Equation of the tangent is obtained by equating to zero the lowest degree terms in the equation (1).
2ay2  xy2 = x3
Equation of tangent:
2ay^{2} = 0 → y^{2} = 0
(v) Asymptote parallel to yaxis: Equation of asymptote is obtained by equating the coefficient of lowest degree of y.
2ay2  xy2 = x3 or (2ax)y2 = x3
Eq. Of asymptote is 2ax = 0 or x = 2a.
(vi) Region of absence of curve: y^{2} becomes negative on putting x>2a or x<0, therefore, the curve does not exist for x<0 and x>2a.
Some more Examples
Example. Trace the witch of Agnesi xy^{2} = a^{2}(ax).
Example. Trace strophoid x (x^{2} + y^{2 }) = a (x^{2 } y^{2} )
Example. Trace the Folium of Descartes x^{3 }+ y^{3 }= 3axy^{
}
Example. Trace y^{2}( a^{2} + x^{2 }) = x^{2}( a^{2 } x^{2}^{ })
