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.
let's intersect them:
x^2 - b = -x^2 + b
2x^2 = 2b
x = ± √b
then y = 0
So the points of intersection are (√b,0) and (-√b,0)
Their vertices are (0,b) and (0,-b)
Which makes the lengths of their diagonals equal to 2b and 2√3
Since the area of a rhombus is the product of their diagonals/2
we have:
2b(2√3)/2 = 54
I will let you finish it.
To find the value of b, we need to find the vertices of the two parabolas and calculate the area of the rhombus formed.
First, let's find the vertices of the parabolas by setting y equal to zero in each equation:
For y = x^2 - b: x^2 - b = 0
For y = -x^2 + b: -x^2 + b = 0
By solving these equations, we can determine the x-coordinate of the vertices. Let's solve the first equation:
x^2 - b = 0
x^2 = b
x = ±√b
Now, let's solve the second equation:
-x^2 + b = 0
x^2 = b
x = ±√b
From these solutions, it can be observed that the x-coordinate of the vertices is the same for both parabolas. Let's denote this x-coordinate as V.
Now, we know the area of the rhombus formed by the intersection points of the parabolas and their vertices is given as 54.
The area of a rhombus is given by the formula: A = (d1 * d2) / 2,
where d1 and d2 are the lengths of the diagonals of the rhombus.
The diagonals of the rhombus are the lines connecting the vertices of the parabolas. As the vertices of the rhombus have equal x-coordinates (V), the diagonal lengths will be the difference in their y-coordinates.
Let's calculate the y-coordinate of the vertices:
For y = x^2 - b, substituting x = ±√b, we get:
y = (√b)^2 - b = b - b = 0
Similarly, for y = -x^2 + b, substituting x = ±√b, we get:
y = - (√b)^2 + b = -b + b = 0
Thus, the vertices of the rhombus formed by the intersection points are (±√b, 0).
Now, let's calculate the diagonal lengths:
d1 = |√b - (-√b)| = 2√b
d2 = |0 - 0| = 0
As mentioned earlier, the area of the rhombus is given by A = (d1 * d2) / 2. Substituting the known values, we get:
54 = (2√b * 0) / 2
54 = 0
Since 54 = 0 is not true, there seems to be an error in the given information or problem statement. Please double-check the question.