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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