Conic Sections Question 105
Question: . Two perpendicular tangents to $ y^{2}=4ax $ always intersect on the line, if
[Karnataka CET 2000]
Options:
A) $ x=a $
B) $ x+a=0 $
C) $ x+2a=0 $
D) $ x+4a=0 $
Show Answer
Answer:
Correct Answer: B
Solution:
We know that tangent to the parabola at points $ t_1 $ and $ t_2 $ are $ t_1y=x+at_1^{2} $ and $ t_2y=x+at_2^{2}. $ Since tangents are perpendicular to the parabola, therefore, $ \frac{1}{t_1}.\frac{1}{t_2}=-1 $ or $ t_1t_2=-1 $ . We also know that their point of intersection $ =(at_1t_2,a(t_1+t_2)) $
$ =(-a,a(t_1+t_2)). $ Thus these points lie on directrix $ x=-a $ or $ x+a=0 $ .
BETA
