derive the equation of the parabolla with its vertex on the line 7x+3y-4=0 and containing ponts (3,-5) and (3/2,1) the axis being horizontal.
Not possible
If the vertex is on the line 7x + 3y - 4 = 0, which has a slope of -7/3,
then its axis would have to have a slope of 3/7, which is NOT horizontal.
Check your question.
the equation is
x = a(y-k)^2 + h
The two points given mean that
3-h = a(-5-k)^2
3/2 - h = a(1-k)^2
Divide to get rid of the a, and we have
(3-h)/(5+k)^2 = (3/2 - h)/(1-k)^2
or
2(17+3k)(1-k)^2 = (13+6k)(5+k)^2
and
k=1
so, h=1
x-1 = a(y+1)
3-1 = a(-5+1)
a = -1/2
finally,
x = -1/2 (y+1)^2 + 1
Misunderstood the question
Go with Steve's answer.
actually, there is a second solution, where k = -97/17, h = 359/119, making a = -1/42
and
x = -1/42 (y + 97/17) + 359/119
(I think)
actually, I botched it near the end:
(h,k) = (1,-1) as solved, but
x-1 = a(y+1)^2
3-1 = a(-4)^2
2 = 16a
a = 1/8
x = 1/8(y+1)^2 + 1
also, the second solution is correctly expressed as
x = -17/504 * (y+97/17)^2 + 359/119
plotting the three graphs shows that the two vertices lie on the line, go through both given points, and open in opposite directions.
*whew*
To derive the equation of a parabola, we need to use the vertex form equation for a parabola in a horizontal axis:
(x - h)^2 = 4p(y - k)
Where (h, k) represents the vertex coordinates, p represents the distance from the vertex to the focus, and (x, y) represents any point on the parabola.
In this case, the vertex lies on the line 7x + 3y - 4 = 0. To find the vertex of the parabola, we need to solve this line equation:
7x + 3y - 4 = 0
Rearranging the equation to isolate y, we have:
3y = -7x + 4
y = (-7/3)x + (4/3)
Now, we can see that the vertex of the parabola lies on this line, and its x-coordinate remains the same. So, the x-coordinate of the vertex is 3.
Now, let's find the y-coordinate of the vertex by substituting the x-coordinate into the equation:
y = (-7/3)(3) + (4/3)
y = -7 + (4/3)
y = -7/3 + 4/3
y = -3/3
y = -1
So, the vertex of the parabola is (3, -1).
Next, we need to find the value of p, which represents the distance from the vertex to the focus. To calculate this, we can use one of the given points on the parabola.
Let's choose the point (3/2, 1). The x-coordinate of this point is half the distance from the vertex to the focus, so it will help us determine the value of p.
3/2 = 3 + p
Rearranging the equation:
p = (3/2) - 3
p = 3/2 - 6/2
p = -3/2
Now, we have the following information:
- Vertex: (3, -1)
- Distance from vertex to focus (p): -3/2
Plugging these values into the vertex form equation, we get:
(x - 3)^2 = 4(-3/2)(y + 1)
Simplifying the equation further:
(x - 3)^2 = -6(y + 1)
And there you have it. The equation of the parabola with a horizontal axis, vertex on the line 7x + 3y - 4 = 0, and passing through the points (3, -5) and (3/2, 1) is:
(x - 3)^2 = -6(y + 1)