A hyperbola for which a = b is called an equilateral hyperbola. Find the eccentricity e of an equilateral hyperbola.
The Correct Answer and Explanation is :
Correct Answer:
The eccentricity ( e ) of an equilateral hyperbola is ( \sqrt{2} ).
Explanation:
An equilateral hyperbola is a type of hyperbola where the lengths of the semi-major axis (a) and semi-minor axis (b) are equal. The standard form of a hyperbola’s equation depends on whether the transverse axis is horizontal or vertical. For an equilateral hyperbola, one of the common forms is:
[ \frac{x^2}{a^2} – \frac{y^2}{a^2} = 1 ]
This can be rewritten by setting ( a^2 = b^2 ) as:
[ x^2 – y^2 = a^2 ]
The eccentricity ( e ) of a hyperbola is defined by the relationship:
[ e = \sqrt{1 + \frac{b^2}{a^2}} ]
Since for an equilateral hyperbola, ( a = b ), substituting ( b = a ) in the eccentricity formula gives:
[ e = \sqrt{1 + \frac{a^2}{a^2}} ]
[ e = \sqrt{1 + 1} ]
[ e = \sqrt{2} ]
Thus, the eccentricity of an equilateral hyperbola is ( \sqrt{2} ). This value indicates that the hyperbola is more “stretched” compared to a circle (which has an eccentricity of 1) but less so compared to parabolas and ellipses with greater eccentricities. The eccentricity reflects how much the conic section deviates from being circular, and in the case of the equilateral hyperbola, it shows a specific geometric symmetry and stretch.
Generating Image of Equilateral Hyperbola:
Let’s create an image illustrating an equilateral hyperbola with its asymptotes and defining axes. The graph will include the hyperbola ( x^2 – y^2 = a^2 ) and its asymptotes ( y = \pm x ), illustrating the properties and symmetry of the equilateral hyperbola.
Here is the image illustrating an equilateral hyperbola, as described. It shows the hyperbola (x^2 – y^2 = a^2) along with its asymptotes (y = x) and (y = -x), depicted in a clear and contrasting manner on a Cartesian coordinate system. The graph visually represents how the curves of the hyperbola approach the asymptotes without intersecting them.