Write the equation of the parabola with a vertex (3,5) and a focus of (3,9)

Answer-

Solution-

Given,

the parabola has vertex at (3,5) and focus at (3,9)

It can be easily noticed that focus and vertex lie on the same vertical line,

Hence, the axis of symmetry is a vertical line ( a line perpendicular to x-axis). Also, the focus lies to the top of the vertex so the parabola will open up upwards.

General form of parabola vertex at (h, k)and vertical line of symmetry is,

Putting the values,

We can find the value of 'a' which is distance between vertex and focus by using distance formula,

Taking only +ve value, as the parabola opens upwards.

then the equation becomes,

Given: Vertex (3,5) and focus (3,9) of the parabola.

As you can see that the x-coordinates of the vertex and focus are the same, so, this is a regular vertical parabola.

The Standard form of the equation of the parabola is; , where a≠0.         ......[1]

The vertex of this parabola is at (h, k).

and the focus of this parabola  is at (h, k+a).

From the given,

h = 3, k=5 and k+a = 9

To solve for a;

k+a = 9

Substitute the value of k =5 in above expression:

5+a =9

⇒ a= 9-5 =4

Therefore, the vertex and focus are 4 units apart.

Hence, the equation of parabola by substituting the value of h, k and a in equation [1] ;

or

.