PQRS is a rectangle such that `PR +PQ = 5PS` and `PR - PS = 8`. Then area (in sq. units) of rectangle is-

Options

A. 40
B. 50
C. 60
D. 70

Answer

To solve the problem, we need to find the area of rectangle PQRS given the conditions:

1. \( PR + PQ = 5 PS \)
2. \( PR - PS = 8 \)

Let's denote:
- \( L \) as the length of the rectangle (PQ or SR)
- \( B \) as the breadth of the rectangle (PS or QR)

### Step 1: Set up the equations based on the problem statement

From the properties of the rectangle:
- \( PR \) (the diagonal) can be expressed using the Pythagorean theorem:
\[ PR = \sqrt{L^2 + B^2} \]
- \( PQ = L \) and \( PS = B \)

Substituting these into the first equation:
\[ \sqrt{L^2 + B^2} + L = 5B \]

From the second equation:
\[ \sqrt{L^2 + B^2} - B = 8 \]

### Step 2: Solve the second equation for \( PR \)

Rearranging the second equation:
\[ \sqrt{L^2 + B^2} = B + 8 \]

### Step 3: Substitute into the first equation

Now substitute \( \sqrt{L^2 + B^2} \) from the second equation into the first equation:
\[ (B + 8) + L = 5B \]
This simplifies to:
\[ L + 8 = 4B \]
Thus, we can express \( L \) in terms of \( B \):
\[ L = 4B - 8 \quad \text{(Equation 1)} \]

### Step 4: Substitute \( L \) back into the second equation

Now we substitute \( L \) back into the expression for \( PR \):
\[ \sqrt{(4B - 8)^2 + B^2} = B + 8 \]

### Step 5: Square both sides to eliminate the square root

Squaring both sides gives:
\[ (4B - 8)^2 + B^2 = (B + 8)^2 \]

Expanding both sides:
\[ (16B^2 - 64B + 64) + B^2 = B^2 + 16B + 64 \]
Combining like terms:
\[ 17B^2 - 64B + 64 = B^2 + 16B + 64 \]

### Step 6: Rearranging the equation

Bringing all terms to one side:
\[ 17B^2 - 64B + 64 - B^2 - 16B - 64 = 0 \]
This simplifies to:
\[ 16B^2 - 80B = 0 \]

### Step 7: Factor the equation

Factoring out \( 16B \):
\[ 16B(B - 5) = 0 \]
Thus, we have two solutions:
1. \( B = 0 \) (not valid since dimensions cannot be zero)
2. \( B = 5 \)

### Step 8: Find \( L \)

Substituting \( B = 5 \) back into Equation 1:
\[ L = 4(5) - 8 = 20 - 8 = 12 \]

### Step 9: Calculate the area of the rectangle

The area \( A \) of the rectangle is given by:
\[ A = L \times B = 12 \times 5 = 60 \text{ square units} \]

### Final Answer

The area of rectangle PQRS is \( \boxed{60} \) square units.

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