A parabola is set of all points in a plane which are an equal distance away from a given point and given line. The point is called the focus of the parabola and the line is called the directrix. The focus lies on the axis of symmetry of the parabola.
The distance between focus and directrix is "a".
Using the formula distance between two points, we get get the value of "a".
Equation of parabola (y - k)2 = 4a(x - h) (y - k)2 = -4a(x - h) (x - h)2 = 4a(y - k) (x - h)2 = -4a(y - k) |
Direction of opening Open rightward Open leftward Open up Open down |
Problem 1 :
Find the equation of parabola with focus (5, 0) and vertex (5, 3).
Solution :
Always foucs and vertex will lie on the same line. The focus is below the vertex, the parabola opens down.
(x - h)2 = -4a(y - k)
Here (h, k) is (5, 3)
(x - 5)2 = -4a(y - 3) ----(1)
To find "a", let us find the distance between focus and vertex,
Distance between two points = √(x2 - x1)2 + (y2 - y1)2
= √(5 - 5)2 + (3 - 0)2
= √9
a = 3
By applying the value of a in (1), we get
(x - 5)2 = -4(3)(y - 3)
(x - 5)2 = -12(y - 3)
Problem 2 :
Find the equation of parabola whose focus is at (6, 3) and the vertex (2, 3) is given by.
Solution :
Always foucs and vertex will lie on the same line. The focus is at next to vertex. So the parabola opens right and it is symmetric about x-axis.
(y - k)2 = 4a(x - h)
Here (h, k) is (2, 3)
(y - 3)2 = 4a(x - 2) ----(1)
To find "a", let us find the distance between focus and vertex,
Distance between two points = √(x2 - x1)2 + (y2 - y1)2
= √(6 - 2)2 + (3 - 3)2
= √42 + 02
a = 4
By applying the value of a in (1), we get
(y - 3)2 = 4(4)(x - 2)
(y - 3)2 = 16(x - 2)
Problem 3 :
Find the equation of parabola whose focus is at (-2, 4) and the vertex (1, 4) is given by.
Solution :
Always foucs and vertex will lie on the same line. The focus is in front of vertex. So the parabola opens left and it is symmetric about x-axis.
(y - k)2 = -4a(x - h)
Here (h, k) is (1, 4)
(y - 4)2 = -4a(x - 1) ----(1)
To find "a", let us find the distance between focus and vertex,
Distance between two points = √(x2 - x1)2 + (y2 - y1)2
= √(1 - (-2))2 + (4 - 4)2
= √32 + 02
a = 3
By applying the value of a in (1), we get
(y - 4)2 = -4(3)(x - 1)
(y - 4)2 = -12(x - 1)
May 21, 24 08:51 PM
May 21, 24 08:51 AM
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