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]
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](https://us-static.z-dn.net/files/d70/67b7ef763807f023def5240f60934b50.jpg)
![Ver imagen merlynthewhizz](https://us-static.z-dn.net/files/d90/bd8982a946c6f07a0f458d0e3f2921c9.jpg)