We will discuss how to find the equation of the parabola whose vertex at a given point and axis is parallel to yaxis.
Let A (h, k) be the vertex of the parabola, AM is the axis of the parabola which is parallel to yaxis. 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 coordinate system at A. Draw two mutually perpendicular straight lines AM and AN through the point A as y and xaxes respectively.
According to the new coordinate axes (x', y ') be the coordinates of P. Therefore, the equation of the parabola is (x’)\(^{2}\) = 4ay' (a > 0) …………….. (i)
Therefore, we get,
AM = y' and PM = x'
Also, OR = k, AR = h, OQ = y, PQ = x
Again, x = PQ
= PM + MQ
= PM + AR
= x' + h
Therefore, x' = x  h
And, y = OQ = OR + RQ
= OR + AM
= k + y'
Therefore, y' = y  k
Now putting the value of x' and y' in (i) we get
(x  h)\(^{2}\) = 4a(y  k), which is the equation of the required parabola.
The equation (x  h)\(^{2}\) = 4a(y  k) represents the equation of a parabola whose coordinate of the vertex is at (h, k), the coordinates of the focus are (h, a + k), the distance between its vertex and focus is a, the equation of directrix is y  k =  a or, y + a = k, the equation of the axis is x = h, the axis is parallel to positive yaxis, the length of its latus rectum = 4a, coordinates of the extremity of the latus rectum are (h + 2a, k + a) and (h  2a, k + a) and the equation of tangent at the vertex is y = k.
Solved example to find the equation of the parabola with its vertex at a given point and axis is parallel to yaxis:
Find the axis, coordinates of vertex and focus, length of latus rectum and the equation of directrix of the parabola x\(^{2}\)  y = 6x  11.
Solution:
The given parabola x\(^{2}\)  y = 6x  11.
⇒ x\(^{2}\)  6x = y  11.
⇒ x\(^{2}\)  6x + 9 = y  11 + 9
⇒ (x  3)\(^{2}\) = y  2
⇒ (x  3)\(^{2}\) = 4 ∙ ¼(y  2) ………….. (i)
Compare the above equation (i) with standard form of parabola (x  h)\(^{2}\) = 4a(y  k), we get, h = 3, k = 2 and a = ¼.
Therefore, the axis of the given parabola is along parallel to positive yaxis and its equation is x = h i.e., x = 3 i.e., x  3 = 0.
The coordinates of its vertex are (h, k) i.e., (3, 2).
The coordinates of its focus are (h, a + k) i.e., (3, ¼ + 2) i.e., (3, \(\frac{9}{4}\)).
The length of its latus rectum = 4a = 4 ∙ ¼ = 1 unit
The equation of its directrix is y + a = k i.e., y + ¼ = 2 i.e., y + ¼  2 = 0 i.e., y  \(\frac{7}{4}\) = 0 i.e., 4y  7 = 0.
11 and 12 Grade Math
From Parabola whose Vertex at a given Point and Axis is Parallel to yaxis to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.