Answer:
[tex]y = \frac{x^2}{2} +5x +\frac{1}{2}[/tex] is the equation of the parabola
Step-by-step explanation:
Let [tex](x_0,y_0)[/tex] be any point on the parabola. Find the distance between [tex](x_0,y_0)[/tex]and the focus. Then find the distance between [tex](x_0,y_0)[/tex] and directrix. Equate these two distance equations and the simplified equation in [tex]x_0[/tex] and [tex]y_0[/tex] is equation of the parabola.
The distance between [tex](x_0,y_0)[/tex] and (2,5) is [tex]\sqrt{ (x_0- (-5))^2+(y_0-1)^2[/tex]
The distance between [tex](x_0,y_0)[/tex] and the directrix, y= -5 is
[tex]|y_0-(-5)| = |y_0 + 5|[/tex] .
Equate the two distance expressions and square on both sides.
[tex]\sqrt{(x_0-(-5))^2+(y_0-1)^2} =|y_0-(-5)|[/tex]
[tex](x_0+5)^2+(y_0-1)^2=(y_0+5)^2[/tex]
Simplify and bring all terms to one side:
[tex]x_0^2 +10x_0 +1-2y_0[/tex] =0
Writing the equation with[tex]y_0[/tex] on one side:
[tex]y_0 = \frac{x_0^2}{2} +5x_0 +\frac{1}{2}[/tex]
This equation in [tex](x_0,y_0)[/tex] is true for all other values on the parabola and hence we can rewrite with (x,y) .
So, the equation of the parabola with focus (-5, 1) and directrix is y= -5 is
[tex]y = \frac{x^2}{2} +5x +\frac{1}{2}[/tex]
