Let we will discuss about

**Cardioide**

**r=a(1+cos theta)**----- If we put

*for*

**-theta***in the equation of the curve we find that;*

**theta**

**r=a{1+cos (-theta)}=a(1+cos theta)**i.e. the equation of the curve does not change. Therefore the given curve is symmetrical about the initial line.

----- Now

*when;*

**r=0**

**1+cos theta = 0**i.e.

**cos theta = -1**therefore

**theta = pi**Hence the curve passes through the origin and the equation of the tangent at the pole is

*i.e. the initial line.*

**theta = pi**----- Now we plot some of the points on the curve.

When

*then*

**theta=0,**

**r=2a;**when

*then*

**theta=pi/3,**

**r=3a/2**when

*then*

**theta=pi/2,**

**r=a**when

*then*

**theta=pi,**

**r=0.**----- From the given equation

*when we have*

**r=a(1+cos theta),**

**(dr/d theta)=-a sin theta**That is, when the value of

*increases from*

**theta***to*

**0***then the value of*

**pi,***decreases and as been stated earlier decreases from*

**r***to*

**2a**

**0**Again, since the given curve is symmetrical about the initial line, therefore when the value of

*increases from*

**theta***to*

**pi**

**2pi,**Then the value of

*increases from*

**r***to*

**0**

**2a.**