Find the equations to the common tangents to the two hyperbolas `(x^2)/(a^2)-(y^2)/(b^2)=1` and `(y^2)/(a^2)-(x^2)/(b^2)=1`

Options

A. (a) The foci of each ellipse always lie within the other ellipse
B. (b) Their auxiliary circles are the same
C. (c) Their director circles are the same
D. (d) The ellipses encloses the same area

Answer

B, C, D

Explanation

Without loss of generality assume `a gt b` Foci of `1^(st)` ellipse are `(+- ae, 0)`
Putting this point in `(x^(2))/(b^(2)) + (y^(2))/(a^(2)) -1`, we get `(a^(2)e^(2))/(b^(2)) -1`
The above quantity may be negative or positive, hence option (a) is not correct
Auxiliary circle for `(x^(2))/(a^(2)) + (y^(2))/(b^(2)) =1` is `x^(2) + y^(2) =a^(2)` for `(x^(2))/(b^(2)) + (y^(2))/(a^(2)) =1` is `x^(2) + y^(2) =a^(2)`
Director circle for `(x^(2))/(b^(2)) +(y^(2))/(a^(2)) =1` is `x^(2)+y^(2) = a^(2) +b^(2)`
`(x^(2))/(b^(2)) +(y^(2))/(a^(2)) =1` is
Area of the ellipse `= pi ab`