Question

Select the equation for a graph that is the set of all points in the plane that are equidistant from the point F(6, 0) and line x = −6.

Answer 1

The equation of the graph is $x = \frac 1{24}y^2$

How to determine the equation?

Generally, a point is represented as:

(x,y)

The line x = -6 means that:

(-6, y)

So, we calculate the distance between (x,y) and (-6, y) as follows:

$d = \sqrt{(x + 6)^2 + (y - y)^2}$

This gives

$d = \sqrt{(x + 6)^2 }$

Next, we calculate the distance between (6,0) and (x, y) as follows:

$d = \sqrt{(x - 6)^2 + (y - 0)^2}$

This gives

$d = \sqrt{(x - 6)^2 + y^2}$

Equate both distance expressions

$\sqrt{(x + 6)^2} = \sqrt{(x - 6)^2 + y^2}$

Square both sides

$(x + 6)^2 = (x - 6)^2 + y^2$

Expand

$x^2 + 12x + 36 = x^2 - 12x + 36 + y^2$

Evaluate the like terms

$12x = - 12x + y^2$

Add 12x to both sides

$y^2 = 24x$

Make x the subject.

$x = \frac 1{24}y^2$

Hence, the equation of the graph is $x = \frac 1{24}y^2$

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