A parabola has a focus of F(2, -0.5) and a directrix of y=-1.5 P(x,y) represents any point on the parabola, while D(x, -1.5) represents any point on the directrix.

What are the steps required to find the equation of the parabola?

Respuesta :

The sketch of the parabola is attached below

We have the focus [tex](a,b) = (2, -0.5)[/tex]
The point [tex]P(x,y)[/tex]
The directrix, c at [tex]y=-1.5[/tex]

The steps to find the equation of the parabola are as follows

Step 1
Find the distance between the focus and the point P using Pythagoras. We have two coordinates; [tex](2, -0.5)[/tex] and [tex](x,y)[/tex].
We need the vertical and horizontal distances to find the hypotenuse (the diagram is shown in the second diagram).
The distance between the focus and point P is given by
[tex] \sqrt{ (x-a)^{2}+ (y-b)^{2} } [/tex]

Step 2
Find the distance between the point P to the directrix [tex]c[/tex]. It is a vertical distance between y and c, expressed as [tex]y-c[/tex]

Step 3
The equation of parabola is then given as 
[tex] \sqrt{ (x-a)^{2}+ (y-b)^{2} } [/tex]=[tex]y-c[/tex]
[tex] (x-a)^{2}+ (y-b)^{2}= (y-c)^{2} [/tex] ⇒ substituting a, b and c
[tex] (x-2)^{2}+ (y--0.5)^{2} = (y--1.5)^{2} [/tex]
[tex] (x-2)^{2}+ (y+0.5)^{2}= (y+1.5)^{2} [/tex]⇒Rearranging and making [tex]y[/tex] the subject gives

[tex]y= \frac{ x^{2} }{2} -2x+1[/tex]

Ver imagen merlynthewhizz
Ver imagen merlynthewhizz
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