The equation `y^(2)-8y-x+19=0` represents

Options

A. a parabola whose focus is `((1)/(4),0)` and directrix is `x=(-1)/(4)`
B. a parabola whose vertex is `(3,4)` and directrix is `x=(11)/(4)`
C. a parabola whose focus is `((13)/(4),4)` and vertex is `(0,0)`
D. a curve which is not a parabola

Answer

B

Explanation

Given equation is
`y^(2)-8y-x+19=0`
`rArr (y-4)^(2)=x-19+16`
`rArr (y-4)^(2)=(x-3) rArr Y^(2)=4Ax`
where, `Y=y-4, A=(1)/(4)`
`and X=x-3`
`therefore " Focus"=(A,0)=((1)/(4),0),`
vertex = (3, 4)
Directrix, `X=-(1)/(4) rArr x-3= -(1)/(4)`
`therefore x=(11)/(4)`