We will discuss how to find the equation of the parabola whose vertex at a given point and axis is parallel to x-axis.
Let A (h, k) be the vertex of the parabola, AM is the axis of the parabola which is parallel to x-axis. The distance between the vertex and focus is AS = a and let P (x, y) be any point on the required parabola.
Now we shift the origin of co-ordinate system at A. Draw two
mutually perpendicular straight lines AM and AN through
the point A as x and y-axes respectively.
According to the new co-ordinate axes (x', y ') be the co-ordinates of P. Therefore, the equation of the parabola is (y')\(^{2}\) = 4ax' (a > 0) …………….. (i)
Therefore, we get,
AM = x' and PM = y'
Also, OR = h, AR = k, OQ = x, PQ = y
Again, y = PQ
= PM + MQ
= PM + AR
= y' + k
Therefore, y' = y - k
And, x = OQ = OR + RQ
= OR + AM
= h + x'
Therefore, x' = x - h
Now putting the value of x' and y' in (i) we get
(y - k)\(^{2}\) = 4a(x - h), which is the equation of the required parabola.
The equation (y - k)\(^{2}\) = 4a(x - h) represents the equation of a parabola whose co-ordinate of the vertex is at (h, k), the co-ordinates of the focus are (a + h, k), the distance between its vertex and focus is a, the equation of directrix is x - h = - a or, x + a = h, the equation of the axis is y = k, the axis is parallel to positive x-axis, the length of its latus rectum = 4a, co-ordinates of the extremity of the latus rectum are (h + a, k + 2a) and (h + a, k - 2a) and the equation of tangent at the vertex is x = h.
Solved example to find the equation of the parabola with its vertex at a given point and axis is parallel to x-axis:
Find the axis, co-ordinates of vertex and focus, length of latus rectum and the equation of directrix of the parabola y\(^{2}\) + 4x + 2y - 11 = 0.
Solution:
The given parabola y\(^{2}\) + 4x + 2y - 11 = 0.
y\(^{2}\) + 4x + 2y - 11 = 0
⇒ y\(^{2}\) + 2y + 1 - 1 + 4x - 11 = 0
⇒ (y + 1)\(^{2}\) = -4x + 12
⇒ {y - (-1)}\(^{2}\) = -4(x - 3)
⇒ {y - (-1)}\(^{2}\) = 4 ∙ (-1) (x - 3) …………..(i)
Compare the above equation (i) with standard form of parabola (y - k)\(^{2}\) = 4a(x - h), we get, h = 3, k = -1 and a = -1.
Therefore, the axis of the given parabola is along parallel to negative x-axis and its equation is y = - 1 i.e., y + 1 = 0.
The co-ordinates of its vertex are (h, k) i.e., (3, -1).
The co-ordinates of its focus are (h + a, k) i.e., (3 - 1, -1) i.e., (2, -1).
The length of its latus rectum = 4 units
The equation of its directrix is x + a = h i.e., x - 1 = 3 i.e., x - 1 - 3 = 0 i.e., x - 4 = 0.
● The Parabola
11 and 12 Grade Math
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