Answer to Question #157470 in Analytic Geometry for Ariel

Question #157470
Write down the standard form of the equation of a parabola in a given directrix and vertex at (0, 0)
(a) x = -4
(b) y = 2
(c) y = - 3
1
Expert's answer
2021-01-29T05:21:34-0500

We can write the equation of parabola in the form:

"4p(y-k)=(x-h)^2" . The focus is (h,k+p) and the directrix is "y=k-p". The vertex is (h,k). In item b) we receive: "k-p=2" and "h=0, k=0". Thus, "p=-2". The equation is: "-8y=x^2"

In c). we get "k-p=-3", "h=0,k=0". The equation is: "12y=x^2"

Now we write the equation in the form: "4p(x-k)=(y-h)^2" . The focus is (h,k+p) and the directrix is "x=k-p". The vertex is (h,k). For a). we receive: "k=0,h=0", "k-p=-4". Thus, "p=4." The equation is: "16x=y^2" . The standard forms of equations of parabola are: "y=ax^2+bx+c" and "x=ey^2+fy+g" , where "a,b,c,e,f,g\\in{\\mathbb{R}}". All obtained equations are in standard forms.

Answer: a). "16x=y^2"; b). "-8y=x^2" ; c). "12y=x^2"



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