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
1.1 General Equation of Conic Section
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
As shown in figure $2$, let us assume the coordinates of focus is $S(\alpha ,\beta )$ and the equation of directrix is $ax + by + c = 0$, then the equation of conic section whose eccentricity is $e$ can be find out using the formulae $$SP = e \times PM$$
Using distance formulae, we get $$\sqrt {{{\left( {x - \alpha } \right)}^2} + {{\left( {y - \beta } \right)}^2}} = e \times \frac{{\left| {ax + by + c} \right|}}{{\sqrt {{a^2} + {b^2}} }}$$
Squaring both sides, we get $${\left( {x - \alpha } \right)^2} + {\left( {y - \beta } \right)^2} = {e^2} \times \frac{{{{(\left| {ax + by + c} \right|)}^2}}}{{{a^2} + {b^2}}}$$
