The area of the rhombus formed by points of intersection of parabolas y=x^2−b and y=−x^2 +b and their vertices is 54. Find b.
The four points are
(0,b),(√b,0),(0,-b),(-√b,0)
Now use your favorite method to find the base length and altitude, or sides and angles, etc. to find the area of the rhombus.
Steve, I am looking for an answer, not a way to find it.
And the area of a rhombus is 54. The question I am asking is Find B, when the equations are y=x^2 -b and y=-x^2 +b
To find the area of the rhombus formed by the points of intersection of the given parabolas, we need to determine the coordinates of the vertices of the rhombus.
First, let's find the points of intersection of the parabolas y = x^2 - b and y = -x^2 + b. To do this, we will set the two equations equal to each other and solve for x:
x^2 - b = -x^2 + b
Combining like terms:
2x^2 = 2b
Dividing both sides by 2:
x^2 = b
Taking the square root of both sides:
x = ±√b
Now that we have the x-values of the points of intersection, we can substitute them back into either of the original equations to find the corresponding y-values:
For y = x^2 - b:
y = (√b)^2 - b
y = b - b
y = 0
For y = -x^2 + b:
y = -(-√b)^2 + b
y = -b + b
y = 0
Thus, the points of intersection are (±√b, 0).
To find the length of one side of the rhombus, we can use the distance formula between two points. Let's take the points (√b, 0) and (-√b, 0):
d = √((x2 - x1)^2 + (y2 - y1)^2)
d = √(((-√b) - (√b))^2 + (0 - 0)^2)
d = √((-2√b)^2 + 0)
d = √(4b)
Since a rhombus has four equal sides, the length of one side is √(4b).
The area of a rhombus can be found by multiplying the lengths of its diagonals and dividing by 2. Since the diagonals of a rhombus are perpendicular bisectors of each other, the length of one diagonal of the rhombus formed by the intersection points is twice the length of one side. Therefore, the length of one diagonal is 2√(4b).
The area of the rhombus is given as 54, so we can set up the equation:
Area = (Diagonal 1 * Diagonal 2) / 2
54 = (2√(4b) * 2√(4b)) / 2
54 = 4b
Simplifying:
4b = 54
b = 54 / 4
b = 13.5
Therefore, b is equal to 13.5.