Answer:
To find the equation of a parabola in standard form given the focus and directrix, we can use the definition of a parabola. A parabola is defined as the set of all points that are equidistant from the focus and the directrix.
In this case, the focus has coordinates (30,0) and the equation of the directrix is x = -30. Let's denote a point on the parabola as (x, y). According to the definition, the distance from this point to the focus is equal to the distance from this point to the directrix.
Using the distance formula, we can calculate these distances. The distance from (x, y) to the focus (30, 0) is given by:
√((x - 30)^2 + (y - 0)^2)
The distance from (x, y) to the directrix x = -30 is simply |x - (-30)| = |x + 30|.
Since these distances are equal, we can set up an equation:
√((x - 30)^2 + y^2) = |x + 30|
To eliminate the square root, we can square both sides of the equation:
(x - 30)^2 + y^2 = (x + 30)^2
Expanding both sides:
x^2 - 60x + 900 + y^2 = x^2 + 60x + 900
Simplifying:
-120x + y^2 = 120x
Rearranging terms:
y^2 = 240x
This is now in standard form, where the equation of the parabola is y^2 = 4px. Comparing this with our equation, we can see that p = 60.
Therefore, the equation in standard form of the parabola with focus (30,0) and directrix x = -30 is y^2 = 240x.
Explanation: