Area Of A Triangle By Heron S Formula

Step 1: Coordinates of the points. The coordinates of point are , and the coordinates of points and are given by the parametric equations of the line:

Step 2: Formula for the area of triangle . The area of triangle is given by: Where and .

Step 3: Calculating the vectors. The vectors and are given by:

Step 4: Using the given area condition. The given area of the triangle is square units, so we have the equation: This implies:

Step 5: Determining the cross product of and . The cross product is calculated as: This determinant simplifies to: Thus:

Step 6: Finding the magnitude. Now, we compute the magnitude of the cross product: We are given that this magnitude is , so: Squaring both sides:

Step 7: Solving for . Solving the above equation leads to the value of , which gives .

Step 1: Understanding the geometry of the region.
We are given the region bounded by the circle , and the line , with . The region described is part of the circle and a sector of the ellipse formed by the line.
Step 2: Finding the area .
The area of the region can be split into two parts:
1. The area of the sector bounded by the circle from to ,
2. The area of the circular region from to , corresponding to the equation .
Thus, the area is given by the integral: Evaluating the first part gives: Step 3: Calculating the desired expression. The expression we need to evaluate is: Substitute the value of from the previous calculation: Simplifying: Thus, the correct value is , and the correct answer is option (1).

To find the maximum area of the triangle with vertices at (0, 0), (x, y), and (−x, y) where , we can proceed as follows:
The base of the triangle is the distance between (x, y) and (−x, y), which is 2x.
The height of the triangle is y, which is the distance from the origin (0, 0) to the line joining (x, y) and (−x, y).
Thus, the area of the triangle is:

Since , we can substitute this into the area expression:

To maximize , we take the derivative with respect to and set it to zero:

Now, substitute back into the equation for :

Thus, the maximum area is:

We are given matrices , , and . We need to calculate the sum of the diagonal elements of matrix .
Step 1: Calculate . First, multiply and : We are given that , and from matrix multiplication rules, we get:
Step 2: Use the formula to calculate the diagonal sum. Through matrix computations, we find the sum of the diagonal elements of is: Thus, .

To find the area of the largest rectangle inscribed within the region , we begin by understanding the boundary conditions that define the region.

1. First, analyze the boundaries:
   • is the line .
   • is a downward-opening parabola intersecting the y-axis at .
   • restricts the region to the positive x-axis.
2. To find the intersection points of the lines, set equal to :
   • .
   • Rearrange: .
3. Use the quadratic formula where , , and :
   • Discriminant .
   • .
   • Thus, possible are (taking the positive square root for a valid x-intersection).
4. Now that we have the boundaries of the region, consider the largest rectangle:
   • The height, .
   • The width can be determined by considering :
   • Therefore, at , maximum possible for largest rectangle occurs when intersects y of parabola: .
5. Calculate maximum area of rectangle:
   • Height , Width .
   • Substitute and compute width:
     
   • Simplifying:
     .

The largest such rectangle has an area of

Since is onto, the range of is .
Now, the derivative , which factors as:
Evaluating at various points:

Hence, the range of is .
Now, for , we have:
Thus, the range of is , and the set . Therefore, the number of elements in is 30.

We are given that . To solve for , we differentiate both sides of the equation with respect to .
Step 1: Differentiate .
First, apply the chain rule: On the right-hand side, differentiate : So, Thus, we find the expression for .
Step 2: Compute .
Using the expression for , we calculate the sum : This sum evaluates to:

We are given the equations and . To find the area of the region bounded by these curves, we first solve for the points of intersection.
Step 1: Solve the system of equations. From the second equation, solve for : Substitute this into the first equation: This simplifies to: Let , so we have: Solve for using the quadratic formula: Thus, or (reject because ). So, , hence .
Step 2: Calculate the area. The area is given by the integral of the difference between the two curves: where and . Calculate the integral and find: