# Director Circle of the Hyperbola

The locus of the point of intersection of the tangents which are at right angles is known as the director circle of the hyperbola.

$$\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{^{2}}}}=1$$.

The equation to the director circle is x² + y² = a² – b².Example: Hyperbola 9x² – 16y² = 144 then find the equation of the director circle.

Solution: The given equation of the hyperbola is 9x² – 16y² = 144

$$\frac{9{{x}^{2}}}{144}-\frac{16{{y}^{2}}}{144}=1$$.

$$\frac{{{x}^{2}}}{16}-\frac{{{y}^{2}}}{9}=1$$.

Where a² = 16

a = 4

b² = 9

b = 3

The equation of the director circle is x² + y² = a² – b².

We can substitute a, b value above formula

x² + y² = 16 – 9

x² +  y² = 7.