I need help with two problems please URGENT

Question

1) A company produces headlights for cars which are parabolic. If the equation for the headlights is x^2= 20y, how far from the vertex should the bulb be placed?

 

2) Many planets and other astronomical bodies have elliptical orbits. One of our most famous comets, Halley’s Comet, has an elliptical orbit with the sun at one focus. The length of the major axis of this orbit is approximately 36.2 AU and the minor axis
length is 9.12 AU. AU stands for ‘Astronomical Unit’ and 1 AU is approximately 93,000,000 miles – the distance from the Earth to the Sun.
Find the eccentricity of the orbit of Halley’s Comet

 

Answer 1

1) Given the equation is 20y = x²

    This is a parabola with the vertex at (0, 0)

 

    Assuming the bulb is placed at the focus of the parabola and the distance between the vertex and the focus is p, then 4p = 20 and p = 5.  The bulb would be 5 units from the vertex.

 

2) If a is the length of the semi-major axis and b is the length of the semi-minor axis, then the eccentricity of the ellipse is e = √(1 - (b/a)²)

 

    The length of the major axis is 36.2 A.U., the length of the semi-major axis is a = 18.1  A.U.

    The length of the minor axis is 9.12 A.U., the length of the semi-minor axis is b = 4.56 A.U.

 

    Putting these values into the equation for eccentricity, e = 0.968