If the normals from any point to the parabola x ^2=4y cut the line y=2 in point whose abscissa are in ap, then the slope of the tangents at the 3 co-normal points are in
Answers
Answered by
0
Step-by-step explanation:
Let (2t,t
2
) is point on the parabola x
2
=4y,
∴ Normal at (2t,t
2
) is t
3
+t(2−y)−x=0 ...(1)
Let three normals passes through (h,k) then t
3
+t(2−k)−h=0
∴t
1
+t
2
+t
3
=0,t
1
t
2
+t
2
t
3
+t
3
t
1
=2−k and t
1
t
2
t
3
=h
Given that, normals cuts the line y=2 then from (1) x=t
3
then abscissaes of three points are t
1
3
,t
2
3
,t
3
3
in A.P.(given)
∴2t
2
3
=t
1
3
+t
3
3
=(t
1
+t
3
)
3
−3t
1
t
3
(t
1
+t
3
)⇒t
2
2
=t
1
t
3
...(2)
∵ Slope of tangents at three co-normal points i.e., t
1
,t
2
,t
3
are in G.P from (2)
Similar questions