Auxiliary circle of a hyperbola is defined as:

The auxiliary circle for a hyperbola is a circle with its centre on the polar and contains the two vertices.

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The circle whose center concurs with that of the ellipse and whose radius is equal to the ellipse's semimajor axis.

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The auxiliary circle for a hyperbola is a circle with its centre on the axis and contains the two vertices.

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None of these

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Solution

The correct option is A The auxiliary circle for a hyperbola is a circle with its centre on the axis and contains the two vertices.
The Auxiliary circle of a hyperbola is defined as the circle with the center same as the hyperbola and with transverse axis as it's diameter. The end points of transverse axis are the two vertices of the hyperbola, so the circle also contains the two vertices of the hyperbola.

The equation of Auxiliary circle foe a hyperbola is given by $x^{2} + y^{2} = a^{2}$

Hence the correct answer is Option $C$ .

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