The equation of the parabola given is:
\[ x^2 = \frac{1}{2}y \]
This can be rewritten in the standard form of a parabola that opens upwards, which is:
\[ x^2 = 4py \]
From the given equation, we can identify \(4p\):
\[ 4p = \frac{1}{2} \implies p = \frac{1}{8} \]
Now, for a parabola of the form \(x^2 = 4py\):
- The focus is located at \((0, p)\).
- The equation of the directrix is given by \(y = -p\).
Using the value of \(p = \frac{1}{8}\):
- The focus \(F\) is at:
\[ F(0, \frac{1}{8}) \]
- The equation of the directrix is:
\[ y = -\frac{1}{8} \]
So the final answer is:
Focus: \(F(0, \frac{1}{8})\); Directrix: \(y = -\frac{1}{8}\)
Thus, the correct response from the options given is:
upper F left parenthesis 0 comma start fraction 1 over 8 end fraction right parenthesis semicolon y equals negative start fraction 1 over 8 end fraction
1 answer