Conic Sections Question 84
Question: If $ y_1,\ y_2 $ are the ordinates of two points P and Q on the parabola and $ y_3 $ is the ordinate of the point of intersection of tangents at P and Q, then
Options:
A) $ y_1,\ y_2,\ y_3 $ are in A.P.
B) $ y_1,\ y_3,\ y_2 $ are in A.P.
C) $ y_1,\ y_2,\ y_3 $ are in G.P.
D) $ y_1,\ y_3,\ y_2 $ are in G.P.
Show Answer
Answer:
Correct Answer: B
Solution:
Let the co-ordinates of P and Q be $ (at_1^{2},2at_1) $ and $ (at_2^{2},2at_2) $ respectively. Then $ y_1=2at_1 $ and $ y_2=2at_2. $ The co-ordinates of the point of intersection of the tangents at $ P $ and Q are $ {at_1t_2,a(t_1+t_2)} $
$ \therefore y_3=a(t_1+t_2) $
therefore $ y_3=\frac{y_1+y_2}{2} $
therefore $ y_1,y_3 $ and $ y_2 $ are in A.P.
BETA
