Match the following lists:

Answer

3

Explanation

Focus of ellipse `-=(kae_(1),0)`
and Focus of hyperbola `-=(ae_(2),0)`
Now, `kae_(1)=ae_(2)`
`rArr" "sqrt(1-(a^(2))/(k^(2)a^(2)))=sqrt(1+(a^(2))/(a^(2)))`
`rArr" "k^(2)-1=2`
`therefore" "k= pmsqrt3`
a. `(e_(2))/(e_(1))=k=sqrt3` ltBrgt b. Major axis of ellipse = 2ka
and
Transverse axis of hyperbola = 2a
`therefore" "(2ka)/(2a)=k=sqrt3`
c. Since curves are confocal, they are orthogonal.
`therefore" "theta=(pi)/(2)`
d. Latus rectum of ellipse and hyperbola are `(2a^(2))/(ak) and (2a^(2))/(a)`, respectively.
`therefore" "(2a//k)/(2a)=(1)/(k)=(1)/(sqrt3)`