Answer to Question #157464 in Analytic Geometry for Tony

Question

Identify the focus and directrix of the following parabola
(a) y^2 = 3x^2

Answer 1

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:

$y = 3x^2$

Then, remembering the normal form of parabola:

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

Reshaping our equation:

$(x - 0)^2 = 4 * 1/4 * 1/3 * (y-0)$

$(x-0)^2=4*1/12(y-0)$

From which:

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

Focus and directrix are:

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

In our case:

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

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

$y^2=3x$

The normal form of parabola in that case:

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

In our case:

$(y-0)^2=4 * 1/4*3x^2$

$(y-0)^2=4*3/4*x^2$

From which:

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

Focus and directrix are:

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

In our case:

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

Answer:

It is not a parabola