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
9.1 Important Result
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
If two tangents on a parabola make an angle of ${90^ \circ }$ with each other, then the intersection points of the tangents lie on the directrix.
Proof: Let us assume the equation of parabola be ${y^2} = 4ax...(1)$
and tangents are drawn at points $P$ and $Q$ intersect at $R$ as shown in figure $20$.
The equation of tangent to the parabola in terms of slope is $$y = mx + \frac{a}{m}$$
$$x{m^2} - ym + a = 0...(2)$$
The two roots of equation $(2)$ are ${m_1}$ and ${m_2}$. Therefore, $${m_1} + {m_2} = \frac{y}{x}$$ and $${m_1} \times {m_2} = \frac{a}{x}$$
Since tangents are perpendicular to each other, $${m_1} \times {m_2} = - 1$$ $$a=-x$$ $$x+a=0$$ which is the equation of directrix.
