Certainly! Let's describe the key features of the given parabola with the equation [tex]\( x^2 = 40y \)[/tex].
### Key features of a parabola with the equation [tex]\( x^2 = 40y \)[/tex]
1. Identification of [tex]\( p \)[/tex]:
- The general form of a parabola that opens upwards or downwards is [tex]\( x^2 = 4py \)[/tex].
- Comparing the given equation [tex]\( x^2 = 40y \)[/tex] with the general form [tex]\( x^2 = 4py \)[/tex], we can identify that [tex]\( 4p = 40 \)[/tex].
- Solving for [tex]\( p \)[/tex], we get [tex]\( p = \frac{40}{4} = 10.0 \)[/tex].
2. Direction in which the parabola opens:
- Since [tex]\( p \)[/tex] is positive ([tex]\( p = 10.0 \)[/tex]), the parabola opens upwards.
3. Coordinates of the focus:
- The focus of the parabola is given by the coordinates [tex]\( (0, p) \)[/tex].
- Substituting the value of [tex]\( p \)[/tex] we found ([tex]\( p = 10.0 \)[/tex]), the coordinates of the focus are [tex]\( (0, 10.0) \)[/tex].
4. Equation for the directrix:
- The directrix of the parabola is a line given by [tex]\( y = -p \)[/tex].
- Substituting the value of [tex]\( p \)[/tex] ([tex]\( p = 10.0 \)[/tex]), the equation for the directrix is [tex]\( y = -10.0 \)[/tex].
### Summary
- The value of [tex]\( p \)[/tex] is [tex]\( 10.0 \)[/tex].
- The parabola opens upwards.
- The coordinates of the focus are [tex]\( (0, 10.0) \)[/tex].
- The equation for the directrix is [tex]\( y = -10.0 \)[/tex].
These are the key features of the parabola described by the equation [tex]\( x^2 = 40y \)[/tex].
