Question #157464
Identify the focus and directrix of the following parabola
(a) y^2 = 3x^2
1
Expert's answer
2021-01-27T06:31:27-0500

Given:

Identify the focus and directrix of the following parabola

(a) y^2 = 3x^2


Solution:


1) It is not a parabola


Possible solutions:


2) If there is a mistake in the left part of the equation and it should be like:

y=3x2y = 3x^2

Then, remembering the normal form of parabola:

(xh)2=4p(yk)(x-h)^2 = 4p(y-k)

Reshaping our equation:

(x0)2=41/41/3(y0)(x - 0)^2 = 4 * 1/4 * 1/3 * (y-0)

(x0)2=41/12(y0)(x-0)^2=4*1/12(y-0)

From which:

h=0;p=1/12;k=0h=0;p=1/12;k=0

Focus and directrix are:

focus=(h,k+p);directrix:y=kpfocus=(h, k+p); directrix:y=k-p

In our case:

focus=(0,1/12);directrix:y=1/12focus=(0, 1/12); directrix: y=-1/12


3) If there is a mistake in the right part of the equation and it should be like:

y2=3xy^2=3x

The normal form of parabola in that case:

(yk)2=4p(xh)(y-k)^2=4p(x-h)

In our case:

(y0)2=41/43x2(y-0)^2=4 * 1/4*3x^2

(y0)2=43/4x2(y-0)^2=4*3/4*x^2

From which:

k=0;p=3/4;h=0k=0;p=3/4;h=0

Focus and directrix are:

focus=(h+p,k);directrix:x=hpfocus=(h+p,k); directrix: x=h-p

In our case:

focus=(3/4,0);directrix:x=3/4focus = (3/4,0); directrix: x=-3/4


Answer:

It is not a parabola


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