Parabola Formulae

Parabola formulae will help us to solve different types of problems on parabola in co-ordinate geometry.

1. In the following standard equations of parabola ‘a’ is the distance between the vertex and focus of the parabola.

(i) When the standard equation of a parabola is y$^{2}$ = 4ax, (a > 0) then, the co-ordinates of vertex are (0, 0), axis of the parabola is along positive x-axis, co-ordinates of the focus are (a, 0), length of latus rectum = 4a and the equation of directrix is x + a = 0.

(ii) When the standard equation of a parabola is y$^{2}$ = -4ax, (a > 0) then, the co-ordinates of vertex are (0, 0), axis of the parabola is along negativex-axis, co-ordinates of the focus are (-a, 0), length of latus rectum = 4a and the equation of directrix is x - a = 0.

(iii) When the standard equation of a parabola is x$^{2}$ = 4ay, (a > 0) then, the co-ordinates of vertex are (0, 0), axis of the parabola is along positive y-axis, co-ordinates of the focus are (0, a), length of latus rectum = 4a and the equation of directrix is y + a = 0.

(iv) When the standard equation of a parabola is x$^{2}$ = -4ay, (a > 0) then, the co-ordinates of vertex are (0, 0), axis of the parabola is along negative y-axis, co-ordinates of the focus are (0, -a), length of latus rectum = 4a and the equation of directrix is y - a = 0.

(v) When the standard equation of a parabola is (y - β)$^{2}$ =  4a (x - α), (a > 0) then, the co-ordinates of vertex are (α, β), axis of the parabola is along parallel to x-axis, co-ordinates of the focus are (a + α, β), length of latus rectum = 4a and the equation of directrix is x + a = α.

(vi) When the standard equation of a parabola is (x - α)$^{2}$ =  4a (y - β), (a > 0) then, the co-ordinates of vertex are (α, β), axis of the parabola is along parallel to y-axis, co-ordinates of the focus are (α, a + β), length of latus rectum = 4a and the equation of directrix is y + a = β.

2. x = ay$^{2}$ + by + c (a  ≠ 0) represents the equation of a parabola whose axis is parallel to x-axis.

3. y = px$^{2}$ + qx + r (p ≠ 0) represents the equation of a parabola whose axis is parallel to y-axis.

4. The position of a point (x$_{1}$, y$_{1}$) with respect to a parabola y$^{2}$ = 4ax (i.e. the point lies outside, on or within the parabola) according as y$_{1}$$^{2}$ - 4ax$_{1}$ >, =, or < 0.

(i) The point P(x$_{1}$, y$_{1}$) lies outside, on or within the parabola y$^{2}$ = -4ax according as y$_{1}$$^{2}$ + 4ax$_{1}$ >, = or <0.

(ii) The point P(x$_{1}$, y$_{1}$) lies outside, on or within the parabola x$^{2}$ = 4ay according as x$_{1}$$^{2}$ - 4ay$_{1}$ >, = or <0.

(ii) The point P(x$_{1}$, y$_{1}$) lies outside, on or within the parabola x$^{2}$ = -4ay according as x$_{1}$$^{2}$ + 4ay$_{1}$ >, = or <0.

5. The best and easiest form to represent the co-ordinates of any point on the parabola y$^{2}$ = 4ax is (at$^{2}$, 2at). Since, for all the values of ‘t’ the coordinates (at$^{2}$, 2at) satisfy the equation of the parabola y$^{2}$ = 4ax.

The equations x = at$^{2}$, y = 2at are called the parametric equations of the parabola y$^{2}$ = 4ax.

● The Parabola

• Concept of Parabola
• Standard Equation of a Parabola
• Standard form of Parabola y22 = - 4ax
• Standard form of Parabola x22 = 4ay
• Standard form of Parabola x22 = -4ay
• Parabola whose Vertex at a given Point and Axis is Parallel to x-axis
• Parabola whose Vertex at a given Point and Axis is Parallel to y-axis
• Position of a Point with respect to a Parabola
  
• Parametric Equations of a Parabola
• Parabola Formulae
• Problems on Parabola

2nd Grade Math Practice 

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