SHORT NOTES: Parabola

Conic Section:

Parabola : e =1

Ellipse : e < 1
Hyperbola : e > 1

Equations of a parabola:

For change in vertex from (0,0) to (h,k) x is replaced by (x-h) and y is replaced by (y-k)

Parametric equations of parabola:

The equation ${y^2} = 4ax$ can be equivalently written in parametric form.

$x = a{t^2},\,y = 2at$

Equation of tangent to $y^2=4ax$ :

Point form at point $\left ( x_1,y_1 \right )$ : $yy_1=2a\left ( x+x_1 \right )$ {Just take derivative}

parametric form : $ty=x+at^2$ at point $\left ( at^2,2at \right )$

Slope form : $y=mx+\frac{a}{m}$ at point $\left ( \frac{a}{m^2},\frac{2a}{m} \right )$

Equation of normal at $\left ( x_1,y_1 \right )$ :

normal is perpendicular to tangent so $m_1m_2=-1$

$y-y_1=-\frac{y_1}{2a}\left ( x-x_1 \right )$

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