Find an equation of the parabola with vertex and focus: vertex (2, 3) and focus (6, 3)
We can make use of the vertex form of the parabola to calculate the equation.
Answer: (y − 3)2 = 16(x − 2) is the equation of the parabola.
Go through the explanation to understand more.
Explanation:
Parabola: The plane curve generated by a point moving so that its distance from a fixed point is equal to its distance from a fixed-line "or" the intersection of a right circular cone with a plane parallel to an element of the cone.
To find the equation of a parabola, using vertex and focus. Let us consider an example: Given, vertex (2, 3) and focus (6, 3)
Whenever vertex (h, k) is given, we must preferably use the vertex form of the parabola:
- (y − k)2 = 4a(x − h) for horizontal parabola
- (x − h)2 = 4a(y − k) for vertical parabola
+ve when the focus is above the vertex (vertical parabola) or when the focus is to the right of the vertex (horizontal parabola)
-ve when the focus is below the vertex (vertical parabola) or when the focus is to the left of vertex (horizontal parabola)
Let us find an equation of the parabola for vertex (2, 3) and focus (6, 3).
It can be observed that both focus and vertex lie on y = 3, thus the axis of symmetry is a horizontal line.
(y − k)2 = 4a(x − h)
a = 6 − 2 = 4 as y coordinates are the same.
Since the focus lies to the left of vertex, a = 4
(y − 3)2 = 4 × 4 × (x − 2)