Topic
Let the parabola (P) has the canonical equation: y2 = 2px (p > 0) and the line x = m (m > 0) cut (P) at two points I, KY distinguish. Prove two points I and KY symmetric about the axis Ox.
Solution method – See details
Step 1: Parameterizing coordinates I, KY straight line PT x = m
Step 2: Change the coordinates I, KY into PT (P) and prove that the coordinates of these two points have opposite signs and then conclude
Detailed explanation
Since \(I,K \in d:x = m\) \(I(m;t),K(m;k)\)
Since \(I,K \in (P)\) \(\left\{ \begin{array}{l}{t^2} = 2pm\\{k^2} = 2pm\end{array} \ right.\)\( \Leftrightarrow {t^2} = {k^2} \Leftrightarrow \left\{ \begin{array}{l}t = k\\t = – k\end{array} \right. \)
With t = k then I and KY duplicate \( \Rightarrow \) t = k unsatisfactory
With t = –k then I(m ; t) and KY(m ; -t). Then I and KY symmetric about the axis Ox (PCM)
