Respuesta :
Answer:
1. A
2. D
3. D
Step-by-step explanation:
The standard form of a parabola is
[tex]y=\frac{1}{4p}(x-h)^2+k[/tex] ..... (1)
Where, (h,k) is vertex, (h,k+p) is focus and y=k-p is directrix.
1. The directrix of a parabola is y=−8 . The focus of the parabola is (−2,−6) .
[tex]k-p=-8[/tex] ...(a)
[tex](h,k+p)=(-2,-6)[/tex]
[tex]k+p=-6[/tex] .... (b)
[tex]h=-2[/tex]
On solving (a) and (b), we get k=-7 and p=1.
Put h=-2, k=-7 and p=1 in equation (1).
[tex]y=\frac{1}{4(1)}(x-(-2))^2+(-7)[/tex]
[tex]y=\frac{1}{4}(x+2)^2-7[/tex]
Therefore option A is correct.
2 The directrix of a parabola is the line y=5 . The focus of the parabola is (2,1) .
[tex]k-p=5[/tex] ...(c)
[tex](h,k+p)=(2,1)[/tex]
[tex]k+p=1[/tex] .... (d)
[tex]h=2[/tex]
On solving (c) and (d), we get k=3 and p=-2.
Put h=2, k=3 and p=-2 in equation (1).
[tex]y=\frac{1}{4(-2)}(x-(2))^2+(3)[/tex]
[tex]y=-\frac{1}{8}(x-2)^2+3[/tex]
Therefore option D is correct.
3. The focus of a parabola is (0,−2) . The directrix of the parabola is the line y=−3 .
[tex]k-p=-3[/tex] ...(e)
[tex](h,k+p)=(0,-2)[/tex]
[tex]k+p=-2[/tex] .... (f)
[tex]h=0[/tex]
On solving (e) and (f), we get k=-2.5 and p=0.5.
Put h=0, k=-2.5 and p=0.5 in equation (1).
[tex]y=\frac{1}{4(0.5)}(x-(0))^2+(-2.5)[/tex]
[tex]y=\frac{1}{2}(x)^2-2.5[/tex]
[tex]y=\frac{1}{2}(x)^2-\frac{5}{2}[/tex]
Therefore option D is correct.
