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Formula Sheets / Maths

Chapter 25

Parabola

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Standard parabola y² = 4ax Tangent, normal and chord

Standard parabola y² = 4ax

focus $(a, 0),$ directrix $x = - a,$ latus rectum $4 a,$ vertex $(0, 0)$
Parametric
$(a t^{2}, 2 a t)$
Focal distance
$S P = x + a$
Focal chord
$t_{1} t_{2} = - 1;$ $\frac{1}{S P} + \frac{1}{S Q} = \frac{1}{a}$

Tangent, normal and chord

Tangent
$y y_{1} = 2 a (x + x_{1});$ $t y = x + a t^{2};$ $y = m x + \frac{a}{m}$
Normal
$y = - t x + 2 a t + a t^{3};$ $y = m x - 2 am - a m^{3}$
Chord through t₁, t₂
$2 x - (t_{1} + t_{2}) y + 2 a t_{1} t_{2} = 0$
Normal meets again
$t_{2} = - t_{1} - \frac{2}{t _{1}}$
Tangents' intersection
$(a t_{1} t_{2}, a (t_{1} + t_{2}))$