Answer:
The standard form of the equation of the parabola with a focus at (0, 2) and a directrix at y = -2 is \(y=\frac{1}{8}x^2\).
Step-by-step explanation:
The standard form of the parabola is
\(y=ax^2+bx+c\)
The general form of the parabola is
\((x-h)^2=4p(y-k)\) ..... (1)
Where, (h,k) is vertex, (h,k+p) is focus and y=k-p is directrix.
It is given that the parabola with a focus at (0, 2) and a directrix at y = -2. It means
\((h,k+p)=(0,2)\)
\(h=0\)
\(k+p=2\) .... (2)
\(y=k-p\Rightarrow k-p=-2\) .... (3)
On solving (2) and (3), we get
\(k=0\)
\(p=2\)
Substitute h=0, k=0 and p=2 in equation (1).
\((x-0)^2=4(2)(y-0)\)
\(x^2=8y\)
Divide both sides by 8.
\(\frac{1}{8}x^2=y\)
Therefore the standard form of the equation of the parabola with a focus at (0, 2) and a directrix at y = -2 is \(y=\frac{1}{8}x^2\).
