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
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