Hyperbola
Hyperbola
nta_abhyas_2025
Grade 11
Question:
Director circle is the set of points from where drawn tangents are perpendicular, in this case $x^2 + y^2 = a^2 - b^2$ (equation of director circle) i.e., $x^2 + y^2 = -9$ is not a real circle, so there is no point from where perpendicular tangents can be drawn.
Step-by-Step Solution
Key Concept: The director circle of a hyperbola exists only when $a^2 - b^2 > 0$; otherwise no perpendicular tangents exist.
The director circle of a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ has equation $x^2 + y^2 = a^2 - b^2$. For the given hyperbola, this gives $x^2 + y^2 = -9$, which is not a real circle since the right side is negative. Therefore, no point exists from which two perpendicular tangents can be drawn to the hyperbola.
Correct Answer: 3