Parabola
1.0 Conic Section
2.0 Parabola
3.0 Standard equation of Parabola
4.0 Focal distance of a point
5.0 General equation of Parabola
6.0 The generalized form of parabola: ${\left( {y - k} \right)^2} = 4a\left( {x - h} \right)$
7.0 Parametric Co-ordinates
7.1 Parametric relation between the coordinates of the ends of a focal chord of parabola
7.2 Important Results
8.0 Equation of tangent to a parabola
9.0 Point of intersection of tangents at any two points on the parabola
10.0 Equation of normal to the parabola
10.1 Point form
10.2 Slope form
10.3 Parametric form
10.4 To find the number of normal drawn from a point to a parabola (CONCEPT THROUGH QUESTIONS):
10.5 Point of intersection of normal at any two points on the parabola
11.0 Relation between parametric coefficients if normal meets parabola
12.0 Important relations
13.0 Circle through co-normal points
14.0 Chord of contact
7.1 Parametric relation between the coordinates of the ends of a focal chord of parabola
7.2 Important Results
10.2 Slope form
10.3 Parametric form
10.4 To find the number of normal drawn from a point to a parabola (CONCEPT THROUGH QUESTIONS):
10.5 Point of intersection of normal at any two points on the parabola
Let ${y^2} = 4ax$ be a parabola, if $PQ$ be a focal chord as shown in figure. Parametric coordinates of point $P$ is $(a{t_1}^2,2a{t_1})$ and point $Q$ is $(a{t_2}^2,2a{t_2})$ with ${t_1}$ and ${t_2}$ as parameter respectively. $PQ$ passes through focus $S(a,0)$.
Therefore, $Q$, $S$ and $P$ are collinear.
Slope of a line joining points $P$ and $S$ $=$ Slope of line joining $Q$ and $S$
$$\frac{{2a{t_1} - 0}}{{a{t_1}^2 - a}} = \frac{{0 - 2a{t_2}}}{{a - a{t_2}^2}}$$ or, $$\frac{{2{t_1}}}{{{t_1}^2 - 1}} = \frac{{2{t_2}}}{{{t_2}^2 - 1}}$$ or, $${t_1}\left( {{t_2}^2 - 1} \right) = {t_2}\left( {{t_1}^2 - 1} \right)$$ $${t_1}{t_2}\left( {{t_2} - {t_1}} \right) + {t_2} - {t_1} = 0$$ $$({t_2} - {t_1})\left( {{t_1}{t_2} + 1} \right) = 0$$
Since, $${t_2} - {t_1} \ne 0$$ Therefore, $${t_1}{t_2} + 1 = 0$$ $${t_1}{t_2} = - 1$$ or, $${t_2} = \frac{{ - 1}}{{{t_1}}}$$
