What is the focus and the directrix of the graph of #x= 1/24y^2#?

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Answer 1 · 2018-02-19T04:07:54

#\text{Focus}\ \ =\ \ (6\ ,\ 0)#
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#\text{Directrix}\ \ \rightarrow\ \ x=-6#
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# #

Parabola Focus:

A parabola is the locus of points such that the distance to a point (the focus) equals the distance to a line (the directrix).

Parabola Standard Equation:

#4p(x-h)=(y-k)^2# is the standard equation for a right-left facing parabola with vertex at #(h, k)#, and a focal length #|p|#.

Rewrite #x=\frac{1}{24y^2}# in the standard form:

#4\cdot 6(x-0)=(y-0)^2#

Therefore,

#(h, k)=(0, 0)# #\ \ \ #and#\ \ \ # #p=6#

The focus of the parabola is represented by #(h, k + p)# and the directrix is #y = k - p#.

Since the parabola is symmetric around the x-axis and so the focus lies a distance #p# from the center #(0, 0)# along the x-axis.

Hence, focus is:#\ \ \ # #(0+p\ ,\ 0)#

#=(0+6\ ,\ 0)#

#=(6\ ,\ 0)#

Parabola is symmetric around the x-axis and so the directrix is a line parallel to the y-axis, a distance #-p# from the center #(0, 0)# x-coordinate,

#x=0-p#

#x=0-6#

#x=-6#