Parabola
9.0 Point of intersection of tangents at any two points on the parabola
9.0 Point of intersection of tangents at any two points on the parabola
Let the equation of parabola be ${y^2} = 4ax$ and two points on parabola be $P(a{t_1}^2,2a{t_1})$ and $Q \equiv \left( {a{t_2}^2,2a{t_2}} \right)$.
Equation of tangents at $P$ and $Q$ are $$y{t_1} = x + a{t_1}^2...(1)$$
and $$y{t_2} = x + a{t_2}^2...(2)$$
respectively. Solving the equations $(1)$ and $(2)$, we get $x = a{t_1}{t_2}$ and $y = a({t_1} + {t_2})$
Therefore, coordinates of point of intersection of two tangents $R(a{t_1}{t_2},a({t_1} + {t_2}))$.
- If $PQ$ is focal chord then ${t_1}{t_2} = - 1$ therefore, coordinates of point of intersection of two tangents $R( - a,a({t_1} + {t_2}))$.
