CHECKING SYMMETIC NATURE OF CURVES

Symmetric nature of curves

Here what does actually symmetric nature mean?

Symmetric nature of the curve mean that it is reflection or

Image  or replica of the curve about some line or point.

Here “about some line or point” usually we consider line x-axis(y=0) or y-axis (x=0)  and the point is origin(0,0)

Now  how to find a curve is symmetric about a some point

1.symmetric about origin

 we replace the y with –y  and the x with –x ,if the equation the curve doesn’t change then the curve is said to be symmetric about origin.

Example: line y=x,y=x³,y=x²ⁿ⁺¹where n is natural number N, etc..

Lets now check y=x³  holds this property good

Given equation y=x³

Replace by y by –y and x by –x, then the equation will become

(-y)=(-x)³

-y=-x³

Y=x³    hence the equation is symmetric about origin.

2.symmtric about some other point(x,y)

How to check the symmetric nature ?

First take the point(x,y) which is told curve is symmetric about

And choose any point which is on the curve and find the image of the point on the curve about (x,y)

If the image point lies on the curve , the curve is said to be symmetric about point(x,y) only when all points on the curve satisfies this property.

Symmetric nature about a line

1.     about  x-axis(y=0)

 we replace y by-y, if there is no change in the equation of cuvre

2.about y-axis(x=0)

We replace the xby –x, if there is no change in the equation .

3.about some other line

For all points on the curve if we find out the image about given line all the images of those points lies on curve .

For checking symmetric take some point on curve and check it out.