# Formulas for Parabola

Formulas for Parabola

Parabola is a Greek word which refers to a particular plane curve. In general words, parabola can also be define as a plane curve of the second degree. Parabola is a curve described by a projectile, moving on a non-resisting medium under the effect of gravity.

The important formulas of parabola are listed below:-

i. Different equations of parabola and some other relations are shown below:

 Equaiton of parabola Vertex Focus Equation of Directrix Axis Length of latus ration $y^2 = 4ax$ (0, 0) (a, 0) X = -a Y = 0 4a $x^2 = 4ay$ (0, 0) (0, a) Y = -a X = 0 4a $(y- k)^2 = 4a(x- h)$ (h, k) (h +a, k) X = h-a Y = k parallel to x-axis 4a $(x- h)^2 = 4a(y- k)$ (h, k + a) Y = k- a X= h parallel to y-axis 4a

ii.    The point $(x_1, y_1)$ lies on parallel $y^2 = 4ax$ if $y_1^2 = 4ax_1$

The point $(x_1, y_1)$ lies out side $y^2 = 4ax$ if $y_1^2 > 4ax_1$

The point $(x_1, y_1)$ lies in side $y^2 = 4ax$ if $y_1^2 < 4ax_1$

iii.    The line y = mx + c will intersect to parabola at two points if $c < \dfrac{a}{m}$ meet the parabola at coincident points if $c = \dfrac{a}{m}$ not cut the parabola if $c > \dfrac{a}{m}$ .

iv.    The point $(a t^2, 2at)$ lies on the parabola $y^2 = 4ax$ for any parameter t.

v.    The equation of tangent to the parabola $y^2 = 4ax$ at $x(x_1, y_1)$ is $yy_1 = 2(x + x_1)$ and tangent to the parabola $x^2 = 4ay$ at $(x_1, y_1)$ is $xx_1 = 2a(y + y_1)$.

vi.    The condition of tangency of a straight line y = mx + c to a parabola $y^2 = 4ax$ is $c = \dfrac{a}{m}$ i.e. the line $y = mx + \dfrac{a}{m}$ is always tangent to the parabola $y^2 = 4ax$.

vii.    Equation of normal at $(x_1, y_1)$ to the parabola $y^2 = 4ax$ is $yy_1 = \dfrac{-y_1}{2a}(x- x_1)$

[ Note: $y = mx- 2am- am^3$ is equation of normal in m form.]

viii.    $SS_1 N= T^2$ are equation of pair of tangents to the parabola $y^2 = 4ax$ drawn from external point $(x_1, y_1)$ where $S = y^2- 4ax, S_1 = y_1^2- 4ax_1, T = yy_1- 2a(x + 4)$

ix.    $yy_1 = 2a(x + x_1)$ represents the equation of chord of contact to the parabola $y^2 = 4ax$ drawn from and external point $(x_1, y_1)$

x.    $yy_1 = 2a(x + x_1)$ represents the polar of the point $(x_1, y_1)$ with respect to the parabola $y^2 = 4ax$

xi.    $\left(\dfrac{n}{l}, \dfrac{-2am}{l} \right)$ is the pole of the line lx + my + x = 0 w.r.to the parabola $y^2 = 4ax$

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