Conic Sections Question 85
Question: The locus of the point of intersection of any two perpendicular tangents to the hyperbola is a circle which is called the director circle of the hyperbola, then the eqn of this circle is
Options:
A) $ x^{2}+y^{2}=a^{2}+b^{2} $
B) $ x^{2}+y^{2}=a^{2}-b^{2} $
C) $ x^{2}+y^{2}=2ab $
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
Equation of hyperbola is $ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 $
Any tangent to hyperbola are $ y=mx\pm \sqrt{a^{2}m^{2}-b^{2}} $
Also tangent perpendicular to this is $ y=\frac{-1}{m}x\pm \sqrt{\frac{a^{2}}{m^{2}}-b^{2}} $
Eliminating m, we get $ x^{2}+y^{2}=a^{2}-b^{2} $ .
BETA
