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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