It is a typical equation of an ellipse in polar form. However, it is easier to identify conic section, its eccentricity, directrix and focus in rectangular coordinates. Hence, let us convert the polar equation in rectangular form.
The relation between polar form (r,theta)(r,θ) and rectangular form (x,y)(x,y) is given by x=rcosthetax=rcosθ and y=rsinthetay=rsinθ i.e. r^2=x^2+y^2r2=x2+y2.
Hence r=0.8/(1-0.8sintheta)r=0.81−0.8sinθ can be written as
r-4/5rsintheta=4/5r−45rsinθ=45 or 5r-4rsintheta=45r−4rsinθ=4
or 5sqrt(x^2+y^2)-4y=45√x2+y2−4y=4
or 25x^2+25y^2=(4+4y)^2=16y^2+32y+1625x2+25y2=(4+4y)2=16y2+32y+16
or 25x^2+9y^2-32y-16=025x2+9y2−32y−16=0
or 25x^2+9(y^2-2xx16/9y+(16/9)^2)-256/9-16=025x2+9(y2−2×169y+(169)2)−2569−16=0
or 25x^2+9(y-16/9)^2=400/925x2+9(y−169)2=4009
or x^2/(400/(9xx25))+(y-16/9)^2/(400/(9xx9))=1x24009×25+(y−169)24009×9=1
or x^2/(4/3)^2+(y-16/9)^2/(20/9)^2=1x2(43)2+(y−169)2(209)2=1
Hence, this is the equation of an ellipse of the form (x-h)^2/a^2+(y-k)^2/b^2=1(x−h)2a2+(y−k)2b2=1,
whose center is (0,16/9)(0,169), major axis parallel to yy-axis is 2xx20/9=40/92×209=409 and minor axis parallel to xx-axis is 2xx4/3=8/32×43=83
eccentricity is given by e=sqrt(1-a^2/b^2)e=√1−a2b2
= sqrt(1-(4/3)^2/(20/9)^2)=sqrt(1-9/25)=0.8
⎷1−(43)2(209)2=√1−925=0.8
Focii are (h,k+-be)(h,k±be) i.e. (0,16/9+-16/9)(0,169±169) i.e. (0,0)(0,0) and (0,32/9)(0,329)
and directrix are y=k+-b/ey=k±be i.e. y=16/9+-25/9y=169±259
i.e. y=41/9y=419 and y=-1y=−1
graph{(25x^2+9y^2-32y-16)(x^2+y^2-0.01)(x^2+(y-32/9)^2-0.01)(y+1)(y-41/9)=0 [-6.31, 6.346, -1.44, 4.884]}
