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.
If you draw out the graphs, you'll see that one of the diagonals of the rhombus is equal to 2b.
Further, you can obtain the points of intersection with the x-axis for the other diagonal.
For y = x^2 − b,
When y = 0, x = +√b/-√b
Hence, one diagonal of the rhombus is 2b, the other is 2√b
Area of a rhombus is given by:
(1/2) * Product of diagonals
= 0.5 * 2b * 2√b
= 2b√b = 54
=> b√b = 27
=> b = 3
To find the area of the rhombus formed by the points of intersection of the parabolas, we first need to determine the vertices of the rhombus.
Step 1: Find the intersection points of the parabolas.
Set the equations of the parabolas equal to each other to find the x-coordinates of the intersection points:
x^2 - b = -x^2 + b
Combine like terms:
2x^2 = 2b
Divide both sides by 2:
x^2 = b
Step 2: Find the y-coordinates of the intersection points.
Substitute the value of x^2 into one of the parabola equations to find y:
y = (x^2) - b
Since x^2 = b, we can substitute b into the equation:
y = b - b
y = 0
So, the intersection points are (±√b, 0).
Step 3: Find the distance between the intersection points.
The distance between the intersection points is equal to twice the value of √b.
Distance = 2 * √b
Step 4: Calculate the area of the rhombus.
The area of a rhombus is given by the formula:
Area = (diagonal1 * diagonal2) / 2
Since the diagonals of a rhombus are perpendicular bisectors, the diagonals of the rhombus formed by the intersection points are twice the value of √b.
Therefore, the diagonals are 2 * √b and 2 * √b.
Substitute the values into the area formula:
Area = (2 * √b * 2 * √b) / 2
Simplify the expression:
Area = (4b) / 2
Area = 2b
Given that the area is 54, we can set up the equation:
2b = 54
Divide both sides by 2:
b = 54 / 2
b = 27
Therefore, the value of b is 27.
To find the value of b, we first need to understand the problem and determine the steps required to solve it.
Let's start by understanding the given information. We have two parabolas with equations:
1. y = x^2 - b
2. y = -x^2 + b
The vertices of these parabolas are obtained when the values inside the parenthesis are zero. Therefore, the vertices of the first parabola are (0, -b), and the vertices of the second parabola are (0, b).
The area of the rhombus formed by the points of intersection of these parabolas can be calculated by finding the length of two diagonals and multiplying them.
Now, let's find the points of intersection. Equating the equations of the parabolas, we have:
x^2 - b = -x^2 + b
Simplifying this equation, we get:
2x^2 = 2b
Dividing both sides by 2, we obtain:
x^2 = b
Taking the square root of both sides, we have:
x = ±√b
Now, substitute the value of x into one of the parabola equations (let's take the first one) to determine the y-values:
y = (√b)^2 - b
y = b - b
y = 0
Therefore, the two points of intersection are (√b, 0) and (-√b, 0).
Next, let's find the lengths of the diagonals of the rhombus. The length of a diagonal can be calculated using the distance formula.
The diagonal connecting the points (√b, 0) and (-√b, 0) is:
D1 = √[(√b - (-√b))^2 + (0 - 0)^2]
D1 = √[(2√b)^2 + 0]
D1 = √(4b)
D1 = 2√b
The diagonal connecting the points (0, -b) and (0, b) is simply 2b.
Now, we can calculate the area of the rhombus by multiplying the lengths of the diagonals:
Area = D1 * D2
54 = (2√b) * (2b)
54 = 4b^2
Dividing both sides by 4, we obtain:
13.5 = b^2
Taking the square root of both sides, we have:
√13.5 = b
Therefore, the value of b is approximately 3.674.
Hence, the value of b that makes the area of the rhombus 54 is approximately 3.674.