If the sum of length of diagonal of a rhombus is `sectheta`, and perimeter is `2tantheta`. Find the area of rhombus.

Answer

To find the area of the rhombus given the conditions in the problem, we can follow these steps:

### Step 1: Understand the Properties of the Rhombus
A rhombus has two diagonals (let's denote them as D1 and D2) that bisect each other at right angles. The area of a rhombus can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times D1 \times D2 \]

### Step 2: Set Up the Equations
From the problem, we know:
1. The sum of the lengths of the diagonals is given as:
\[ D1 + D2 = \sec \theta \]
2. The perimeter of the rhombus is given as:
\[ \text{Perimeter} = 4a = 2 \tan \theta \]
From this, we can find the length of one side (a):
\[ a = \frac{2 \tan \theta}{4} = \frac{\tan \theta}{2} \]

### Step 3: Relate the Diagonals to the Side Length
Using the relationship between the diagonals and the side length of the rhombus, we have:
\[ D1^2 + D2^2 = 4a^2 \]
Substituting \(a\):
\[ D1^2 + D2^2 = 4 \left(\frac{\tan \theta}{2}\right)^2 = \tan^2 \theta \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \(D1 + D2 = \sec \theta\)
2. \(D1^2 + D2^2 = \tan^2 \theta\)

Let \(D1 = x\) and \(D2 = y\). We can express \(y\) in terms of \(x\):
\[ y = \sec \theta - x \]

Substituting this into the second equation:
\[ x^2 + (\sec \theta - x)^2 = \tan^2 \theta \]

Expanding and simplifying:
\[ x^2 + (\sec^2 \theta - 2x \sec \theta + x^2) = \tan^2 \theta \]
\[ 2x^2 - 2x \sec \theta + \sec^2 \theta = \tan^2 \theta \]

### Step 5: Use the Identity \(\tan^2 \theta + 1 = \sec^2 \theta\)
Substituting \(\tan^2 \theta\):
\[ 2x^2 - 2x \sec \theta + \sec^2 \theta = \sec^2 \theta - 1 \]
\[ 2x^2 - 2x \sec \theta + 1 = 0 \]

### Step 6: Solve the Quadratic Equation
Using the quadratic formula:
\[ x = \frac{-(-2\sec \theta) \pm \sqrt{(-2\sec \theta)^2 - 4 \cdot 2 \cdot 1}}{2 \cdot 2} \]
\[ x = \frac{2\sec \theta \pm \sqrt{4\sec^2 \theta - 8}}{4} \]
\[ x = \frac{\sec \theta \pm \sqrt{\sec^2 \theta - 2}}{2} \]

### Step 7: Find D1 and D2
Using \(D1 = x\) and \(D2 = \sec \theta - x\), we can find both diagonals.

### Step 8: Calculate the Area
Finally, substitute \(D1\) and \(D2\) into the area formula:
\[ \text{Area} = \frac{1}{2} \times D1 \times D2 \]

After evaluating, we find that the area of the rhombus is:
\[ \text{Area} = \frac{1}{4} \]