If the chord joining the points t₁ and t₂ to the parabola y² = 4ax is normal to the parabola at ​t₁ then prove that t₁(t₁ + t₂) = -2.

Dear Student,
$$\begin{gathered} Given, \\ y2 = 4ax, \\ will give \\ \frac{ dy}{dx} = \frac{2a}{y}. \\ Hence, \\ slope of \tan gent at t_1 is\frac{ 2a}{2a{t}_1} = \frac{1}{t_1}. \\ So slope of normal at point t_1 is -t_1. \\ Also slope of line joining t_1 and t_2 is \left(\frac{2a{t}_2 - 2a{t}_1}{a{t_2}^2 - a{t_1}^2}\right) , \\ =\left(\frac{2a(t_2-t2)}{a(t_2-t_1)(t_2+t_1)}\right) \\ =\left(\frac{ 2}{t_1+t_2}\right). \\ The two slopes are the same. \\ Hence \frac{ 2}{t_1+t_2} = -t_1, \\ t_1(t_1+t_2) = -2. \end{gathered}$$



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