| 1. Symmetry
(i) A curve is symmetrical about x-axis if the equation remains the same by replacing y by
–y. here y should have even powers only.
For example y2 = 4ax.
(ii) It is symmetrical about y-axis if it contains only even powers of x For example x2 = 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 x3 + 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 = x3
2. (a) Curve through Origin
The curve passes through the origin, if the equation does not contain constant term.
2 (b)Tangents at the origin:
For example the curve y2 = 4ax passes through 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 co-ordinates 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 y-axis 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. y2 = 4x
Example. a2x2 = y3(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 x-axis is obtained by equating to zero, the coefficient of the highest power of x.
For example yx2 - 4x2 + x + 2 = 0
(y - 4)x2 + 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 y-axis is obtained to zero, the coefficient of highest power of y.
xy3 - 2y3 + y2 + x2 + 2 = 0
(x-2)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 y-axis.
Put 0=dx/dy for the points where tangent is parallel to the x-axis.
For example x2 + y2 - 4x + 4y -1 = 0 ………(1)
2x +2ydxdy- 4 + 4dxdy= 0
(2y+4) dx/dy= 4 - 2x or dx/dy=4224+−yx
Now dxdy= 0. 4-2x = 0 or x = 2
Putting x =2 in (1), we get y2 + 4y -5 = 0
y= 1,- 5
The tangents are parallel to x-axis at the points (2,1) and (2, -5).
Prepare a table for certain values of x and y and draw the curve passing through them.
For example y2 = 4x +4
Example. Trace the curve y2( 2a – x ) = x3
Solution: y2= x3/( 2a – x ) ……..(1)
(i) Origin: Equation does not contain any constant term; therefore, it passes through origin.
(ii) Symmetrical about x-axis: Equation contains only even powers of y; therefore, it is symmetrical about x-axis.
(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:
2ay2 = 0 → y2 = 0
(v) Asymptote parallel to y-axis: Equation of asymptote is obtained by equating the coefficient of lowest degree of y.
(vi) Region of absence of curve: y2 becomes negative on putting x>2a or x<0, therefore, the curve does not exist for x<0 and x>2a.
2ay2 - xy2 = x3 or (2a-x)y2 = x3
Eq. Of asymptote is 2a-x = 0 or x = 2a.
Some more Examples
Example. Trace the witch of Agnesi xy2 = a2(a-x).
Example. Trace strophoid x (x2 + y2 ) = a (x2 - y2 )
Example. Trace the Folium of Descartes x3 + y3 = 3axy
Example. Trace y2( a2 + x2 ) = x2( a2 - x2 )