Parabola formulae will help us to solve different types of problems on parabola in coordinate 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 coordinates of vertex are (0, 0), axis of the parabola is along positive xaxis, coordinates 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 coordinates of vertex are (0, 0), axis of the parabola is along negativexaxis, coordinates 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 coordinates of vertex are (0, 0), axis of the parabola is along positive yaxis, coordinates 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 coordinates of vertex are (0, 0), axis of the parabola is along negative yaxis, coordinates 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 coordinates of vertex are (α, β), axis of the parabola is along parallel to xaxis, coordinates 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 coordinates of vertex are (α, β), axis of the parabola is along parallel to yaxis, coordinates 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 xaxis.
3. y = px\(^{2}\) + qx + r (p ≠ 0) represents the equation of a parabola whose axis is parallel to yaxis.
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 coordinates 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.
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