r = a sec q \ r =

\ r cos q = a

\ x = a which is a straight line parallel to y - axis at a distance 'a' units from the origin.

r = a (1 + cos q) = r = a + a cos q

\ r. r = ar + ar cos q or r^{2} - ar cos q = ar

then (r^{2} - ar cos q)^{2} = a^{2}r^{2}

\ [x^{2} + y^{2} - ax]^{2} = a^{2} [x^{2} + y^{2}]

This is a cardioid.

r = a (1 + sin q ) \ r = a + a sin q

\ r . r = a.r + a r sin q

\ r^{2} - ar sin q = ar

\ (r^{2}- ar sin q)^{2} = a^{2} r2

\ [x^{2} + y^{2} - ay]^{2} = a^{2} [x^{2} + y^{2} ] which is cordioid

r =

\ 2r - r cos q = 4

\ 2r = 4 + r cos q

\ (2r)^{2} = ( 4 + r cos q)^{2} \ 4r^{2} = ( 4 + r cos q )^{2}

\ 4 ( x^{2} + y^{2} ) = ( 4 + x )^{2} or
3x^{2 }+ 4y^{2} - 8x -16 = 0

which is the rectangular form of an ellipse.