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** **From ****Parabola Formulae**** to HOME PAGE**

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