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Describe the key features of a parabola with the equation [tex]x^2 = 40y[/tex].

1. The value of [tex]p[/tex] is [tex]10[/tex].
2. The parabola opens [tex] \square [/tex].
3. The coordinates of the focus are [tex] \square [/tex].
4. The equation for the directrix is [tex] \square [/tex].

Sagot :

GinnyAnswer
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].
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