To describe the key features of the parabola given by the equation [tex]\(x^2 = 40y\)[/tex], we proceed as follows:
### The value of [tex]\(p\)[/tex]
1. Form of the equation: The given equation [tex]\(x^2 = 40y\)[/tex] can be compared with the standard form of a parabola that opens upwards, which is [tex]\(x^2 = 4py\)[/tex].
2. Comparing coefficients: By comparing [tex]\(x^2 = 40y\)[/tex] with [tex]\(x^2 = 4py\)[/tex], we identify that [tex]\(4p = 40\)[/tex].
3. Solving for [tex]\(p\)[/tex]: Dividing both sides by 4, we find that [tex]\(p = 10\)[/tex].
So, the value of [tex]\(p\)[/tex] is [tex]\(10\)[/tex].
### The direction in which the parabola opens
Since the equation is [tex]\(x^2 = 40y\)[/tex], and the coefficient of [tex]\(y\)[/tex] is positive, it indicates that the parabola opens upwards.
### The coordinates of the focus
1. Standard focus: For a parabola with the equation [tex]\(x^2 = 4py\)[/tex], the focus is at [tex]\((0, p)\)[/tex].
2. Using [tex]\(p\)[/tex]: Given [tex]\(p = 10\)[/tex], the focus is at the point [tex]\((0, 10)\)[/tex].
So, the coordinates of the focus are [tex]\((0, 10)\)[/tex].
### The equation of the directrix
1. Directrix derivation: The directrix of a parabola [tex]\(x^2 = 4py\)[/tex] is given by [tex]\(y = -p\)[/tex].
2. Using [tex]\(p\)[/tex]: Given [tex]\(p = 10\)[/tex], the directrix is [tex]\(y = -10\)[/tex].
So, the equation of the directrix is [tex]\(y = -10\)[/tex].
Summarizing the key features:
- The value of [tex]\(p\)[/tex]: [tex]\(10\)[/tex]
- The parabola opens: Upwards
- The coordinates of the focus: [tex]\((0, 10)\)[/tex]
- The equation for the directrix: [tex]\(y = -10\)[/tex]
