Plss... help me.. what is the equation of parabola with vertex on the line y=x, axis parallel to Ox, and passing through (6,-2) and (3,4)? thanks!
Since the axis is horizontal, the general equation is
x = a(y-k)^2 + h
Since the vertex is at y=k, k = h and that means
x = a(y-k)^2 + k
Using the two points, we have
a(-2-k)^2 + k = 6
a(4-k)^2 + k = 3
solve those, and you wind up with
x = 1/4 (y-2)^2 + 2
x = 5/36 (y-14/5) + 14/5
so, there are two parabolas which meet the conditions.
See the graphs at
http://www.wolframalpha.com/input/?i=plot+x+%3D+1%2F4+(y-2)%5E2+%2B+2,+x+%3D+5%2F36+(y-14%2F5)%5E2+%2B+14%2F5,+y%3Dx
vertex at point (v,v)
axis parallel to x, opening right
(y-v)^2 = 4a(x-v)
so
(-2-v)^2 = 4 a (6-v)
and
( 4-v)^2 = 4 a (3-v)
======================now plug and chug
4 + 4v + v^2 = 24 a - 4 a v
16- 8v + v^2 = 12 a - 4 a v
-----------------------------
-12 +12v = 12 a or a = v-1
4 + 4 v + v^2=24(v-1)-4v(v-1)
v^2+4v+4 = 24v -24 -4v^2 +4 v
5v^2 -24 v +28 = 0
(5v-14)(v-2)=0
v = 2 or v = 14/5
if v = 2
a = 1
(y-2)^2= 4(1)(x-2)
y^2 -4 y + 4 = 4 x -8
4x = y^2 -4y + 12 check (it works)
I will let you try v = 14/5 :)
To find the equation of a parabola with the given conditions, we can use the general equation of a parabola:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
Since the vertex is on the line y = x, we know that the x-coordinate and y-coordinate of the vertex are the same. Therefore, h = k.
Let's find the equation step by step:
Step 1: Calculate the x-coordinate of the vertex using the fact that the vertex lies on the line y = x.
Since h = k, we have h = x-coordinate of the vertex = y-coordinate of the vertex.
Therefore, we need to solve the equation h = y = x.
Step 2: Once we have the x-coordinate of the vertex, we can plug it into one of the given points to find the value of a.
Step 3: Substitute the values of h, k, and a in the general equation of the parabola to obtain the final equation.
Let's calculate the equation now:
Step 1: Since the vertex lies on the line y = x, the x-coordinate of the vertex is x = h = k.
Step 2: Plug the x-coordinate of the vertex into one of the given points, (6, -2) or (3, 4). Let's take (3, 4) for this example.
x = 3, y = 4:
4 = a(3 - h)^2
Step 3: Substitute the values of h = k and x = 3 to solve for a.
4 = a(3 - h)^2
4 = a(3 - h)(3 - h)
4 = a(9 - 6h + h^2)
4 = 9a - 6ah + ah^2
ah^2 - 6ah + (9a - 4) = 0
Now, we have obtained a quadratic equation in terms of a. We can solve this equation to find the value of a.
Using the fact that the parabola passes through (6, -2):
x = 6, y = -2:
-2 = a(6 - h)^2
Substitute x = 6 and y = -2 into the equation:
-2 = a(6 - h)^2
-2 = a(36 - 12h + h^2)
ah^2 - 12ah + (36a + 2) = 0
Now, we have two equations in terms of h and a. We can solve this system of equations to find the values of h and a.
Once we find the values of h and a, we can substitute them into the general equation y = a(x - h)^2 + h to obtain the final equation of the parabola.