Answer:
Step-by-step explanation:
r=24/(4+2sin θ)
4r+2r sin θ=24
divide by 2
2r+r sin θ=12
2√(x²+y²)+y=12
2√(x²+y²)=12-y
squaring
4(x²+y²)=144-24y+y²
4x²+3y²+24y=144
4x²+3(y²+8y+16-16)=144
4x²+3(y+4)²-48=144
4x²+3(y+4)²=144+48
4x²+3(y+4)²=192
divide by 192
[tex]\frac{x^2}{48}+\frac{(y+4)^2}{64}=1[/tex]
which is an ellipse with centre(0,-4) major axis on y-axis.
[tex]a^2=64\\a=8\\major axis=2*8=16\\b^2=48\\b=4\sqrt{3}\\minor ~axis=2*4 \sqrt{3}=8 \sqrt{3}\\b^2=a^2(1-e^2)\\48=64(1-e^2)\\1-e^2=48/64=3/4\\e^2=1-3/4=1/4\\e=1/2\\c^2=a^2-b^2\\c^2=64-48=16\\c=4[/tex]
