The conclusion about the values of e for rlliptical equations. The provided equations of conic sections are, A) frac{x^{2}}{36} - frac{y^{2}}{13}=1 B)

Concept used: The general equation of an ellipse with center ${\displaystyle (0,0)}$ is, ${\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1.}$ The value of e for the above equation is ${\displaystyle e=\frac{c}{d}}$ Consider the equations of conic sections,

A) ${\displaystyle \frac{x^2}{36}-\frac{y^2}{13}=1}$

B) ${\displaystyle \frac{x^2}{4}-\frac{y^2}{4}=1}$

C) ${\displaystyle \frac{x^2}{25}-\frac{y^2}{16}=1}$

D) ${\displaystyle \frac{x^2}{25}-\frac{y^2}{39}=1}$

E) ${\displaystyle \frac{x^2}{17}-\frac{y^2}{81}=1}$

F) ${\displaystyle \frac{x^2}{36}-\frac{y^2}{36}=1}$

G) ${\displaystyle \frac{x^2}{16}-\frac{y^2}{65}=1}$

H) ${\displaystyle \frac{x^2}{144}-\frac{y^2}{140}=1}$

On comparing the above equations with the general form of an ellipse, the elliptical equations are (C), (E) and (H). The value of eccentricity e for the conic section of the equation ${\displaystyle \frac{x^2}{25}+\frac{y^2}{16}=1\text{is}e=0.6.}$ The value of eccentricity e for the conic section of the equation ${\displaystyle \frac{x^2}{17}+\frac{y^2}{81}=1\text{is}e=0.89.}$ The value of eccentricity e for the conic section of the equation ${\displaystyle \frac{x^2}{144}+\frac{y^2}{140}=1\text{is}e=0.33.}$ Therefore, the value of e for elliptical equations lies between 0 and 1 i.e. ${\displaystyle 0<e<1.}$