NIMCET 2012 — Mathematics PYQ

Let the equation of the standard hyperbola be:

The eccentricity $e$ is given by:

Taking the reciprocal:

2. Eccentricity of the Conjugate Hyperbola ($e^{\prime }$):

The equation of the conjugate hyperbola is:

The eccentricity $e^{\prime }$ is given by:

Taking the reciprocal:

3. Summing the Reciprocals:

Conclusion:

The sum of the reciprocals of the squares of the eccentricities of a hyperbola and its conjugate is always $1$.

Correct Option: (b)