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 the parabola y^2 = 4ax

Related posts:

  1. Co-ordinate Geometry Formulas Co-ordinate Geometry Formulas co-ordinate geometry is one of the most...
  2. Vector Formulas Vector Formulas A vector can also be defined as an...
  3. Ratio and Proportion Formulas As you know Ratio is a relation between two quantities...
  4. Distance Formula Basic Distance Formula. Distance formula to calculate distance between two...
  5. Measurement of Angles Formulas Measurement of Angles Formulas The concept of angle is one...