Help in Analytic geometry problem finding the equation of the locus?

Hi,

The locus is the parabola formed around the directrix x - 4 = 0 or x = 4, and the focus point of (-4,-5).

The vertex would be halfway between them at (0,-5). To go from the vertex to the focus you go 4 squares to the left, so a = -4.

The parabola is x - 0 = -4(y + 5)² or x = -4(y² + 10y + 25) or x = -4y² - 40y - 100 <==ANSWER

I hope that helps!! :-)

Draw a diagram of the situation.

Basically, every point in the locus is going to be an equal distance from the point, (-4, -5) as from the nearest point on the line, x - 4 = 0.

Let's call the locus point (x, y).

Using the distance formula, the distance from this unknown point to the one given is:

D = √[ (x - (-4))^2 + (y - (-5))^2 ]

D = √[ (x + 4)^2 + (y + 5)^2 ]

D = √[ (x + 4)^2 + (y + 5)^2 ]

The distance from the locus to the line is a little trickier. Of course, it always helps to have a picture of the situation, but if we rewrite the equation of the line:

x - 4 = 0

x = 4

Then we can write the coordinates of the nearest point on the line as:

(4, y)

Then the distance, which is equal to distance D, is given by:

D = √[ (x - 4)^2 + (y - y)^2 ]

D = √[ (x - 4)^2 ]

Then we set both distances equal to each other:

√[ (x + 4)^2 + (y + 5)^2 ] = √[ (x - 4)^2 ]

(x + 4)^2 + (y + 5)^2 = (x - 4)^2

x^2 + 8x + 16 + (y + 5)^2 = x^2 - 8x + 16

16x + (y + 5)^2 = 0

(y + 5)^2 = -16x

Your answer is a parabola, given in standard form.

The point and line that we used to find the equation of the locus are called the focus and directrix of the parabola, respectively.

~~~