Question

Find the values of e, a, b, and c for r = 24/(4 + 2sin theta)

Answer 1

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

$\frac{x^2}{48}+\frac{(y+4)^2}{64}=1$

which is an ellipse with centre(0,-4) major axis on y-axis.

$a^2=64\\a=8\\major axis=2*8=16\\b^2=48\\b=4\sqrt{3}\\minor ~axis=2*4 \sqrt{3}=8 \sqrt{3}\\b^2=a^2(1-e^2)\\48=64(1-e^2)\\1-e^2=48/64=3/4\\e^2=1-3/4=1/4\\e=1/2\\c^2=a^2-b^2\\c^2=64-48=16\\c=4$