A parabola has a vertex at (0,0); a horizontal axis; and the point (12,6) is on the parabola. Write the equation of the parabola in standard form.
the equation must of the form
x = ay^2
sub in the point (12,6) to find a
x^2=9/144y
To write the equation of a parabola in standard form, you need to use the general equation for a parabola with a horizontal axis of symmetry. The standard form of a parabola with the vertex (h, k) is given by:
(x – h)^2 = 4p(y – k)
where "p" is the distance between the vertex and the focus/directrix. Since the vertex of the given parabola is at (0,0) and it has a horizontal axis of symmetry, the equation becomes:
x^2 = 4p(y – 0)
Simplifying further, we have:
x^2 = 4py
Now, we need to determine the value of "p" using the information that the point (12,6) lies on the parabola.
Substitute the coordinates (x = 12, y = 6) into the equation:
(12)^2 = 4p(6)
144 = 24p
Now, solve for "p" by dividing both sides of the equation by 24:
p = 144/24
p = 6
Finally, substitute the value of "p" back into the equation:
x^2 = 4(6)y
Simplifying further:
x^2 = 24y
Therefore, the equation of the given parabola in standard form is x^2 = 24y.