The circle with major axis as diameter is called the auxiliary circle of the hyperbola.
If $a > b,$ then the equation of auxiliary circle is

x^{2} + y^{2} = a^{2}

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x^{2} + y^{2} = b^{2}

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x^{2} + y^{2} = a^{2} - b^{2}

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x^{2} + y^{2} = a^{2} + b^{2}

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Solution

The correct option is B $x^{2} + y^{2} = a^{2}$
For any Hyperbola of the form $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$

Length of Major axis is always $2 a$ , though $a$ can be greater, lesser or equal to $b$

For any conjugate Hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = - 1$ or $\frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} = 1$

The length of major axis is always $2 b$ , Though $b$ can be greater, lesser or equal to $a$

So For any Hyperbola the circle drawn taken major axis as a diameter also called auxiliary circle of the Hyperbola will have a diameter of $2 a$ , equal to the length of major axis and center same as center of Hyperbola.

Hence equation of Auxiliary circle of any Hyperbola will be $x^{2} + y^{2} = a^{2}$

So the correct option is $A$

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The circle with major axis as diameter is called the auxiliary circle of the hyperbola. If a>b, then the equation of auxiliary circle is

The circle with major axis as diameter is called the auxiliary circle of the hyperbola.
If $a > b,$ then the equation of auxiliary circle is