We will discuss about the standard equation of a parabola.
Let S be the focus and the straight line ZZ', the directrix of the required parabola. Let SK be the straight line through S perpendicular to the directrix, bisect SK at A and K being the point of intersection with the directrix.
Then
AS = AK
⇒ Distance of A from the focus = Distance of A from the directrix
⇒ A lies on the parabola
Let SK = 2a, where, a > 0.
Then AS = AK = a.
If this line SK intersects the parabola at A then SK is the axis and A is the vertex of the parabola. Draw the straight line AY through A perpendicular to the axis. Now, we choose the origin of coordinates at A and x and yaxis along AS and AY respectively.
Let P (x, y) be any point on the required parabola. Join SP and draw PM and PN perpendicular to the directrix ZZ' and xaxis. Then,
PM = NK = AN + AK = x + a
Now, P lies on the parabola ⇒ SP = PM
⇒ SP\(^{2}\) = PM\(^{2}\)
⇒ (x – a)\(^{2}\) + (y – 0)\(^{2}\) = (x + a)\(^{2}\)
⇒ y\(^{2}\) = 4ax, which is the required equation of the parabola. The equation of a parabola in the form y\(^{2}\) = 4ax is known as the standard equation of a parabola.
Notes:
(i) The parabola has two real foci situated on its axis one of which is the focus S and the other lies at infinity. The corresponding directrix is also at infinity.
(ii) The vertex of the parabola y\(^{2}\) = 4ax is at the origin i.e., the coordinates of its vertex are (0, 0).
(iii) The coordinates of the focus S of the parabola y\(^{2}\) = 4ax are (a, 0).
(iv) The axis of the parabola y\(^{2}\) = 4ax is positive xaxis (assuming a> 0).
(v) The parabola is
symmetric with respect to with respect to its axis. If the point P(x, y) lies on the parabola y\(^{2}\) = 4ax
with respect to xaxis, then the point Q (x, y) also lies on it.
(vi) We have, y\(^{2}\) = 0 when x = 0; hence, the straight line x = 0 (i.e., yaxis) intersects the parabola y\(^{2}\) = 4ax at coincident points. Therefore, yaxis is a tangent to the parabola y\(^{2}\) = 4ax at the origin.
(vii) The line segment PQ is the double ordinate of P and PQ = 2y.
(viii) The coordinates of the end points of the latus rectum L\(_{1}\)L\(_{2}\) of the parabola y\(^{2}\) = 4ax are (a, 2a) and (a, 2a) respectively
(ix) The length of the latus rectum of the parabola y\(^{2}\) = 4ax is 4a.
(ix) The equation of the directrix of the parabola y\(^{2}\) = 4ax is x =  a ⇒ x + a = 0.
(x) The directrix of the parabola y\(^{2}\) = 4ax is parallel to yaxis and it passes through the point K ( a, 0).
(xi) x = at\(^{2}\), y = 2at is the parametric form of the parabola y\(^{2}\) = 4ax and t is called the parameter.
(xii) The coordinates of any point on the parabola y\(^{2}\) = 4ax can be represented as (at\(^{2}\), 2at) where (at\(^{2}\), 2at) are called the parametric coordinates of a point on the parabola y\(^{2}\) = 4ax.
(xiii) From the standard equation of the parabola y\(^{2}\) = 4ax we see that the value of y becomes imaginary when x < 0. Therefore, no portion of the parabola y\(^{2}\) = 4ax lies to the left of yaxis.
Again, if x is positive and gradually increases then y also increases and for each positive value of x we get two values of y which are equal and opposite in signs. Therefore, the curve extends to infinity on the right of the yaxis.
11 and 12 Grade Math
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