Question

Find the vertex ,axis,focus,directrix , LR of the parabola y2 -8y -x +19=0

Answer 1

Answer: Vertex= (3,4), axis= x-axis, focus= (13/4,4), directrix- x=11/4, and LR= 1 unit

Step-by-step explanation:

Given that,

y^2-8y-x+19=0

Rearranging the above equation of parabola to separate x and y, we get:

y^2-8y+16= x-3

(y-4)^2= x-3

Now, equating the above equation with the standard form of parabola, (y-k)^2=4a(x-h), we get:

h=3, k=4, a=1/4 where (h,k) is the coordinate of vertex and a is the focal length.

The axis of the parabola is x axis because the parabola is symmetric about x axis. The parabola has its face opened toward positive x-axis.

So,

vertex= (3,4)

axis= x-axis

focus= (3+1/4,4)=(13/4,4)

directrix, x= 3-1/4=11/4

Latus rectum, LR= 4a= 4x1/4=1