Question

Find an equation of a parabola with a vertex at the origin and directrix X = 2.5

Answer 1

Vertex=(0,0)
directrix is x=2.5
or x-2.5=0
so focus  S is (-2.5,0)
let P(x,y)be any point on the parabola.
Let M be the foot of perpendicular from P(x,y) on the directrix.
then SP=PM
$\sqrt{(x-(-2.5))^2+(y-0)^2} =| \frac{x-2.5}{( \sqrt{1})^2 } | squaring both sides$$(x+2.5)^2+y^2=(x-2.5)^2 , x^2+6.25+5x+y^2=x^2+6.25-5x$
$y^2=x^2+6.25-5x-x^2-6.25-5x=-10x so eq. is y^{2} =-10x$