Question

Find the equation of the ellipse that has its foci at (2,1) and (2,-7) and b= 2

Answer 1

Foci: (2,1) and (2,-7)

`F_1=\text{ (0+ x', 0 + y')}`

The center of the ellipse is 2, 3

`\begin{gathered} F_1=\text{ (0+ 2, 1 + 3)}=>F_1(0,4) \\ F_2=(0+2,-7+3)\Rightarrow F_2(0,-4) \\ \text{Hence, c=4} \end{gathered}`

`F_2=\text{ (0+x', 0 + y')}`

From

`\begin{gathered} c^2=a^2-b^2 \\ a^2=c^2+b^2 \\ c=4,\text{ b=2} \\ a^2=4^2+2^2=16+4=20 \\ a^2=20 \end{gathered}`

The general equation of an ellipse with center ( x' ,y')

where, x'=2 and y' =3

`((x-x^(\prime))^2)/(b^2)+((y-y^(\prime))^2)/(a^2)=1`
`\begin{gathered} ((x-2)^2)/(4)+((y-3)^2)/(20)=1 \\ (5(x-2)^2+(y-3)^2)/(20)=1 \end{gathered}`
`\begin{gathered} 5(x-2)^2+(y-3)^2=20 \\ 5(x^2-4x+4)+(y^2-6y+9)=20 \\ 5x^2-20x+20+y^2-6y+9=20 \\ 5x^2+y^2-20x-6y+29-20=0 \\ 5x^2+y^2-20x-6y+9=0 \end{gathered}`

Hence the equation of the ellipse is

`5x^2+y^2-20x-6y+9=0`