Question

The reciprocal of the eccentricity of a rectangular hyperbola is
(A) 2
(B) $\dfrac{1}{2}$
(C) $\surd2$
(D) $\dfrac{1}{\surd2}$

Answer 1

Hint: The set of all points in a plane whose difference in distance from two fixed points, referred to as foci, in the plane is a constant is known as a hyperbola. The ratio of the distance from the centre to one of the foci (c) and the distance from the centre to one of the vertices makes up the eccentricity of a hyperbola (a).



Formula Used: The eccentricity of the hyperbola is given by the formula
e= $\dfrac{c}{a}$
Where c = distance of centre to one of the foci
And a = distance of centre from one of the vertices



Complete step by step solution:A hyperbola is referred to as a rectangular hyperbola if the length of the transverse axis (2a) in the hyperbola equals the length of the conjugate axis (2b).
Now, $c^2=a^2+b^2$
As 2a = 2b, so, a = b
Hence, $c^2=a^2+a^2=2a^2$
Thus,$ c=\surd2a$
The eccentricity of the rectangular hyperbola is e= $\dfrac{c}{a}$ = $\dfrac{\surd2a}{a}$ = $\surd2$


Thus, The reciprocal of the eccentricity of rectangular hyperbola is $\dfrac{1}{\surd2}$.



Option ‘D’ is correct



Note: As is well known, the distance from the centre to the foci (c) in a hyperbola is greater than or equal to the distance from the centre to one of the vertices. Thus, the eccentricity of a hyperbola is always greater than or equal to one.