The vertex of a parabola is halfway between the focus and directrix. Here, that is at ((h+p)/2, k) so the focal length is (h+p)/2- p= (h-p)/2. Since the "standard" parabola, with horizontal axis, is [itex]4d(x- x_0)= (y- y_0)^2[/itex], here that would be [4(h-p)/2](x- (h+p)/2)= (y- k)^2 which can be written as [itex]x= \frac{1}{2(h-p)}(y- k)^2+ \frac{h+p}{2}[/itex]. If you take that "(h+p)/2" inside the parentheses you get exactly what you have. Well done!you_of_eh said:For a parabola whose Directrix is given by the equation x=p and whose Focus is (h,k).
Is this by any chance the correct general form of the parabola?
x=1/2(h-p) [y^2 - 2yk + h^2+k^2-p^2]
The general form of a parabola is y = ax^2 + bx + c, where a, b, and c are constants and x is the independent variable.
The general form of a parabola allows for any value of a, b, and c, while the standard form is written as y = a(x-h)^2 + k where (h,k) is the vertex of the parabola and a determines the shape and direction of the parabola.
To graph a parabola in general form, you can use the following steps:
x = -b/2a and substituting it into the equation to find the corresponding y value.x and solving for y.a in the general form of a parabola?The constant a in the general form of a parabola determines the direction and shape of the parabola. If a is positive, the parabola opens upwards and if a is negative, the parabola opens downwards. The absolute value of a also determines the width of the parabola, with a larger absolute value resulting in a narrower parabola.
The general form of a parabola is often used in physics to model the trajectory of objects under the influence of gravity. It is also used in engineering and architecture to design structures such as bridges and arches. In business, it can be used to model profit or cost functions. In general, the parabola is a useful tool for understanding and predicting the behavior of quadratic relationships in various fields.