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
7.2 Important Results
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
1. Harmonic mean of the segment of the focal chord is equal to the length of semi-latus rectum i.e., $$\frac{{2 \times SP \times SQ}}{{SP + SQ}} = 2a$$
Proof: Let us assume that the equation of parabola be ${y^2} = 4ax$. $PQ$ be the focal chord with coordinates $P(a{t_1}^2,2a{t_1})$ and $Q(a{t_2}^2,2a{t_2})$. The coordinates of focus is $S(a,0)$.
Apply distance formulae between point $P$ and $S$, we get $$SP = \sqrt {{{\left( {a{t_1}^2 - a} \right)}^2} + {{\left( {2a{t_1} - 0} \right)}^2}} = \sqrt {{a^2}{t_1}^4 + {a^2} - 2{a^2}{t_1}^2 + 4{a^2}{t_1}^2} = a{t_1}^2 + a$$
Similarly, $$SQ = a{t_2}^2 + a$$
So, the harmonic mean of the segment of the focal chord is
$$\frac{{2 \times SP \times SQ}}{{SP + SQ}} = \frac{{2 \times \left( {a{t_1}^2 + a} \right) \times \left( {a{t_2}^2 + a} \right)}}{{\left( {a{t_1}^2 + a} \right) + \left( {a{t_2}^2 + a} \right)}}$$ $$ = \frac{{2{a^2}\left( {{t_1}^2 + 1} \right)\left( {{t_2}^2 + 1} \right)}}{{a\left( {{t_1}^2 + 1 + {t_2}^2 + 1} \right)}}$$ $$ = \frac{{2a\left( {{t_1}^2 + {t_2}^2 + 2{{({t_1}{t_2})}^2}} \right)}}{{\left( {{t_1}^2 + 1 + {t_2}^2 + 1} \right)}}$$
From parametric relation between the coordinates of the ends of a focal chord, we get $${t_1}{t_2} = - 1$$
Therefore, $$\frac{{2 \times SP \times SQ}}{{SP + SQ}} = 2a$$
2. If a focal chord makes an angle with the axis of parabola then the length of focal chord is $$4a{\bf{cose}}{{\bf{c}}^2}\alpha $$
Proof: Let us assume that the end points of a focal chord which makes an angle $\alpha $ with the axis of parabola be $P(a{t_1}^2,2a{t_1})$ and $Q(a{t_2}^2,2a{t_2})$ $$\tan \alpha = slope{\text{ }}of{\text{ }}PQ$$ $$ = \frac{{2a{t_2} - 2a{t_1}}}{{a{t_2}^2 - a{t_1}^2}} = \frac{2}{{{t_2} + {t_1}}}$$ or, $${t_2} + {t_1} = 2\cot \alpha ...(1)$$
Length of focal chord using distance formulae is $$PQ = a{\left( {{t_2} - {t_1}} \right)^2}$$ $$ = a\left[ {{{\left( {{t_2} + {t_1}} \right)}^2} - 4{t_1}{t_2}} \right]{\text{ }}\left( {\because {t_1}{t_2} = - 1} \right)$$ $$ = a\left[ {{{\left( {{t_2} + {t_1}} \right)}^2} + 4} \right]$$
Put the value of ${t_2} + {t_1}$ from equation $(1)$, we get $$PQ = a\left( {4{{\cot }^2}\alpha + 4} \right)$$ $$ = 4a{\text{cose}}{{\text{c}}^2}\alpha $$
