Find the equation of the director circle of the hyperbola $4x^2-3y^2=12$

Question

A very easy question. The director circle is of the form $x^2+y^2=a^2-b^2$

So $$x^2+y^2=3-4=-1$$

Does this mean the circle doesn’t exist? I don’t understand the implications of this result. I checked the graph for the hyperbola, but found no reason why it cannot exist. Why can’t the circle exist?

Answer 1

The circle cannot exist because the maximum angle possible between any two tangents of the hyberbola is strictly less than 90 degrees.

A director circle is the locus of all points where the tangents are perpendicular. Hence the result.

Answer 2

Yes, the circle does not exist as there is no point from where perpendicular tangents can be drawn to the given hyperbola. This happens because in a hyperbola there is no compulsion for a>b or b>a.