Conic Sections Question 108
Question: The tangent drawn at any point P to the parabola $ y^{2}=4ax $ meets the directrix at the point K, then the angle which KP subtends at its focus is
[RPET 1996, 2002]
Options:
A) 30o
B) 45o
C) 60o
D) 90o
Show Answer
Answer:
Correct Answer: D
Solution:
Here, $ P(at^{2},2at) $ and S(a, 0). If the tangent at P, $ ty=x+at^{2}, $ meets the directrix $ x=-aatk, $ then $ k=( -a,\frac{at^{2}-a}{t} ) $
$ m_1= $ slope of $ SP=\frac{2at}{a(t^{2}-1)} $
$ m_2= $ slope of $ SK=\frac{a(t^{2}-1)}{-2at} $
Clearly $ m_1m_2=-1 $ ,
$ \therefore \angle PSK=90^{o}. $
BETA
