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  • If are three points on a parabola <img alt="\mathrm{y^2=4 a x }" src="https://ent

If \mathrm{P, Q, R} are three points on a parabola \mathrm{y^2=4 a x }such that whose ordinates of \mathrm{P, Q, R } are in geometrical progression, then the tangents at \mathrm{P } and \mathrm{R} meet on
 

Option: 1

 the line through Q parallel to x-axis
 


Option: 2

 the line through Q parallel to y-axis
 


Option: 3

 the line joining Q to the vertex
 


Option: 4

 the line joining Q to the focus


Answers (1)

best_answer

Let the coordinates of \mathrm{P, Q, R} be \mathrm{\left(a t_1^2, 2 a t_i\right) i=1,2,3} respectively such that \mathrm{t_1, t_2, t_3} are in G.P. i.e. \mathrm{t_1 t_3=t_2^2}. Equations of the tangents at P and R are
\mathrm{t_1 y=x+a t_1^2}  and \mathrm{t_3 y=x+a t_3^2}, which intersect at the point
\mathrm{\frac{x+a t_1^2}{t_1}=\frac{x+a t_3^2}{t_3} \Rightarrow x=a t_1 t_3=a t_2^2 }
which is a line through Q parallel to y-axis.

Posted by

seema garhwal

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