Question

If $y_1$ and $y_2$ are the ordinates of two points $P$ and $Q$ on the parabola and $y_3$ is the ordinate of the points of intersection of tangents at $P$ and $Q,$ then

• 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 $G.P.$

Answer 1

The correct option is B $y_1,y_3,y_2$ are in $A.P.$

Let the coordinates of $P$ and $Q$ be $({at}_1^2,{2at}_1)$ and $({at}_2^2,{2at}_2)$ respectively.

Then $y_1=2a{t}_1$ and $y_2={2at}_2.$

The coordinates of the point of intersection of the tangents at $P$ and $Q$ are $({at}_1{t}_2,a(t_1+t_2)).$

$\therefore y_3=a(t_1+t_2)\Rightarrow y_3=\frac{y_1+y_2}{2}\Rightarrow y_1,y_2,y_3$ are in $A.P.$