Conic-Sections Question 569
Question: If the tangents at P and Q on a parabola meet in T, Then SP, ST and SQ are in
Options:
A) A.P
B) GP.
C) H.P.
D) None of these
Show Answer
Answer:
Correct Answer: B
Solution:
[b] Let $ P(at^2_1,2at_1) $ and $ Q(at^2_2,2at_2) $ to two points on the parabola $ y^{2}=4ax. $ The tangents at P and Q intersect at $ T[at_1t_2,a(t_1+t_2)]. $ Now, $ SP=\sqrt{{{(at_1^{2},-a)}^{2}}+2{{(at_1-0)}^{2}}} $ $ =a(t^2_1+1); $ $ SQ=a(t^2_2+1) $ and $ ST=\sqrt{{{(at_1t_2-a)}^{2}}+{{[a(t_1+t_2)-0]}^{2}}} $ $ =a\sqrt{(1+t^2_1)(1+t^2_2)} $
$ \Rightarrow ST^{2}=a^{2}(1+t^2_1)(1+t^2_2)=SP.SQ $ Hence SP.ST, SQ are in G.P.
BETA
