Question

Question #89f79

Answer 1

See below.

Explanation:

`f(x,y) = (a_1 x + b_1 y + c_1)^2 + (a_2 x + b_2 y + c_2)^2 - 1=0` or

`f(x,y) = 1/2 p. H.p + C.p + d`

where

`p = {x,y}`
`H = ((a_1^2 + a_2^2, a_1 b_1 + a_2 b_2),(a_1 b_1 + a_2 b_2, b_1^2 + b_2^2))`
`C =2 (a_1c_1+a_2c_2, b_1c_1+b_2c_2)`
`d= c_1^2+c_2^2-1`

Now `f(x,y)` represents a general conicoid. To have internal finite area, this conicoid should verify

Roots of `det(H - lambda I_2) = 0` ( `H` eigenvalues) should be both positive or both negative. or

`lambda_1 = a_1^2 + a_2^2 + b_1^2 + b_2^2 - sqrt[((a_2 + b_1)^2 + (a_1 - b_2)^2) ((a_2 - b_1)^2 + (a_1 + b_2)^2)] ne 0`
`lambda_2 = a_1^2 + a_2^2 + b_1^2 + b_2^2 + sqrt[((a_2 + b_1)^2 + (a_1 - b_2)^2) ((a_2 - b_1)^2 + (a_1 + b_2)^2)] ne 0`
`lambda_1 lambda_2 > 0`