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
8.2 Tangent in terms of slope
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 the parabola be $${y^2} = 4ax...(1)$$
and the given straight line is $$y=mx+c...(2)$$
Put value of $y$ from equation $(2)$ in equation $(1)$, we get $${\left( {mx + c} \right)^2} = 4ax$$ $${m^2}{x^2} + {c^2} + 2mcx = 4ax$$ $${m^2}{x^2} + 2x\left( {mc - 2a} \right) + {c^2} = 0$$
If Discriminant $D>0$, then line cuts the parabola.
If Discriminant $D=0$, then the line is tangent to the parabola.
If Discriminant $D<0$, then line never touches or cuts the parabola.
So the condition for tangent is $D=0$, i.e., $$4{\left( {mc - 2a} \right)^2} - 4{m^2}{c^2} = 0$$ $$4{m^2}{c^2} + 16{a^2} - 16amc - 4{m^2}{c^2} = 0$$ $$16{a^2} = 16amc$$ $$c = \frac{a}{m}$$
So, the equation of tangent to the parabola is $$y = mx + \frac{a}{m}$$
Similarly for the parabola ${x^2} = 4ay$, the equation of tangent is $$x = my + \frac{a}{m}$$
To find point of contact when slope of tangent on a parabola is given
Let the equation of parabola be $${y^2} = 4ax...(1)$$
and the equation of tangent in terms of slope can be written as $$y = mx + \frac{a}{m}...(2)$$
Differentiate equation $(1)$ with respect to $x$, we get $$2y\frac{{dy}}{{dx}} = 4a$$ $$\frac{{dy}}{{dx}} = \frac{{2a}}{y}$$
Therefore, slope of tangent $m$ at point $P({x_1},{y_1})$ is $${\frac{{dy}}{{dx}}_{\left( {{x_1},{y_1}} \right)}} = \frac{{2a}}{{{y_1}}} = m$$
or, $${y_1} = \frac{{2a}}{m}$$
Now, put value of ${y_1}$ in equation $(2)$, we get $${y_1} = m{x_1} + \frac{a}{m}$$ $$\frac{{2a}}{m} = m{x_1} + \frac{a}{m}$$ $${x_1} = \frac{a}{{{m^2}}}$$
Hence, the point of contact is $(\frac{a}{{{m^2}}},\frac{{2a}}{m})$ where $m \ne 0$. This is also called as $m-$ point on the parabola.
