Conic Sections
Conic Section
Allen Star Batch
Grade 11
Question:
For which of the following hyperbolas, we can have more than one pair of perpendicular tangents?
$\frac{x^2}{4} - \frac{y^2}{9} = 1$
$\frac{x^2}{4} - \frac{y^2}{9} = -1$
$x^2 - y^2 = 4$
$xy = 44$
Step-by-Step Solution
Key Concept: The director circle is a standard result for any conic section, obtained from the condition that two tangents are perpendicular.
The locus of intersection points of perpendicular tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is the director circle $x^2+y^2=a^2+b^2$, which exists as a real circle only when $a>b$.
Correct Answer: 2