Find an equation of a parabola with a vertex at the origin and directrix y=-1.5
Since the vertex is at the origin and the directrix is a horizontal line, the parabola must be opening upward.
First, let's find the focus of the parabola. The focus is also on the axis of symmetry, which is the vertical line x=0. It is a distance p=1.5 units above the vertex. Therefore, the focus is at (0, 1.5).
Next, we use the distance formula to write an equation that relates the distance from any point on the parabola to both the vertex and the focus:
d = sqrt(x^2 + y^2) # distance to vertex
d' = sqrt((x-0)^2 + (y-1.5)^2) # distance to focus
Since the directrix is y=-1.5, the distance from any point on the parabola to the directrix is simply the distance from that point to the line y=1.5. Therefore, we can set these two distances equal to each other to get our equation:
d = d' - 1.5
Squaring both sides and simplifying, we get:
x^2 + y^2 = (y-1.5)^2 + 2.25
x^2 + y^2 = y^2 - 3y + 2.25 + 2.25
x^2 = -3y + 4.5
This is the equation of the parabola in standard form. We can also write it in vertex form by completing the square:
x^2 = -3y + 4.5 # isolate x^2
x^2 = -3(y - 1.5) + 4.5 # factor out -3 from y - 1.5
x^2 = -3(y - 1.5)^2 + 6.75 # complete the square
Therefore, the equation of the parabola is:
x^2 = -3(y - 1.5)^2 + 6.75
identify the center and radius of a circle with equation (x-4)^2+(y+2)^2=9
The center of the circle is located at the point (4, -2), which is the opposite of the signs inside the parentheses. This is because the standard form equation of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.
To find the radius, we simply take the square root of the constant term on the right side of the equation. In this case, the constant term is 9, so:
r = sqrt(9) = 3
Therefore, the center of the circle is (4, -2) and the radius is 3.
AAAaannndd the bot gets it wrong yet again!
what a lot of work for the parabola!
and you knew from the start that since the vertex is at (0,0) the equation will be of the form y = ax^2
we know that the parabola x^2 = 4py has its directrix as y = -p
So, that makes it easy to get the equation:
x^2 = 6y
or
y = 1/6 x^2
I apologize for the mistake in my previous response. You are correct that since the vertex is at (0, 0), the equation of the parabola will be in the form y = ax^2.
To find the value of a, we can use the fact that the directrix is y = -1.5, which is a distance of 1.5 units below the vertex. The distance from the vertex to the focus is also 1.5 units. Therefore, we have:
a = 1/4p = 1/(4*1.5) = 1/6
So the equation of the parabola is:
y = 1/6 x^2
Thank you for bringing this to my attention and please let me know if you have any other questions!
To find an equation of a parabola with a vertex at the origin and a given directrix, we can use the standard form equation of a parabola: (y - k)^2 = 4p(x - h), where (h, k) represents the coordinates of the vertex and p represents the distance between the vertex and the focus/directrix.
In this case, the vertex is at (0, 0) since the vertex is at the origin. And the directrix is y = -1.5.
Since the directrix is a horizontal line, the equation should be in terms of x. The distance between the vertex and the directrix is equal to the value of p.
The distance between the vertex and the directrix is 1.5 units. Therefore, p = 1.5.
Plugging these values into the standard form equation, we have: y^2 = 4(1.5)x
Simplifying, we get the equation of the parabola: y^2 = 6x