Question

For the hyperbola $\frac{x^2}{49}-\frac{y^2}{25}=1$, the equation of auxillary circle is

• A. $x^2+y^2=49$
• B. $x^2+y^2=25$
• C. $x^2+y^2=10$
• D. $x^2+y^2=10074$

Answer 1

Answer: (A)

$x^2+y^2=49$

For any Hyperbola of the form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$,

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

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

Here the given hyperbola is $\frac{x^2}{49}-\frac{y^2}{25}=1$, which is similar to standard form of the hyperbola.

Here $a=7$ and $b=5$

The equation of Auxiliary circle for the given hyperbola will also be $x^2+y^2=(7{)}^2$

$\Rightarrow x^2+y^2=49$

So the correct option is $A$