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(yk)=(xh)24p(y-k)=(x-h)^2 . The focus is (h,k+p) and the directrix is y=kpy=k-p. The vertex is (h,k). In item b) we receive: kp=2k-p=2 and h=0,k=0h=0, k=0. Thus, p=2p=-2. The equation is: 8y=x2-8y=x^2

In c). we get kp=3k-p=-3, h=0,k=0h=0,k=0. The equation is: 12y=x212y=x^2

Now we write the equation in the form: 4p(xk)=(yh)24p(x-k)=(y-h)^2 . The focus is (h,k+p) and the directrix is x=kpx=k-p. The vertex is (h,k). For a). we receive: k=0,h=0k=0,h=0, kp=4k-p=-4. Thus, p=4.p=4. The equation is: 16x=y216x=y^2 . The standard forms of equations of parabola are: y=ax2+bx+cy=ax^2+bx+c and x=ey2+fy+gx=ey^2+fy+g , where a,b,c,e,f,gRa,b,c,e,f,g\in{\mathbb{R}}. All obtained equations are in standard forms.

Answer: a). 16x=y216x=y^2; b). 8y=x2-8y=x^2 ; c). 12y=x212y=x^2



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