Solve Lesson 70 Page 97 SBT Math 10 – Kite>


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)

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