3.2
Equations of the parabola
We want to deduce the Cartesian
equation of the ellipse from its locus definition. You have already seen
an example in the matric question in the previous section. But here we
want to use the properties of the focus and the directrix in our equation.
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Derive
the Cartesian equation for the parabola from its focal definition
– assume that the vertex is at the origin, and that
the focus has coordinates (0, p).
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The standard equation of a parabola with vertex at the origin
and vertical orientation is 4py
= x2,
where p is the distance between the vertex and the origin.
When the
vertex is not at the origin, but at the point (h, k), the
standard form of the equation of the parabola is 4p(y
– k) = (x
– h)2.
The standard equation of a parabola with vertex at the origin
and horizontal orientation is 4px
= y2,
where p is the distance between the vertex and the origin.
When the vertex is not at the origin, but at the point (h,
k), the standard form of the equation of the parabola is 4p(x
– h) = (y
– k)2.
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We can use this form
to better understand our traditional school Cartesian parabola equation
y
– k = a(x
– h)2. In this equation, we all know the physical meaning
of h and k, but what is the meaning of a? We know that if a is positive
the arms are “up” and if it is negative the arms are downward.
Also, as the value of a increases, the arms become narrower.
But what physical meaning do we have for a?
- The
focal form if the equation of the parabola 4py
= x2
is equivalent to the Cartesian form
y = ax2.
Prove that the width of this parabola through the focus
is 4p, and deduce a physical meaning for a.
- Find
the equation of the parabola if the vertex is at the origin
and the focus has
co-ordinates (0, -6).
- Find
the co-ordinates of the focus given the equation y2
= 12x.
- Suppose
you are given the equation y
= -6x2.
How can you find the focus and directrix of this parabola?
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