Two circles of radii 4 cm and 9 cm respectively touch each other externally at a point and a common tangent touches them at the points P and Q respectively. Then the area of a square with one side PQ, is

Options

A. 97 sq.cm
B. 194 sq.cm
C. 72 sq.cm
D. 144 sq.cm

Answer

To solve the problem, we need to find the area of a square whose side length is equal to the length of the common tangent (PQ) between two circles with given radii. Here’s a step-by-step solution:

### Step 1: Identify the radii of the circles
Let the radius of the first circle (r1) be 4 cm and the radius of the second circle (r2) be 9 cm.

### Step 2: Calculate the distance between the centers of the circles
Since the circles touch each other externally, the distance (d) between their centers is the sum of their radii:
\[ d = r1 + r2 = 4 \, \text{cm} + 9 \, \text{cm} = 13 \, \text{cm} \]

### Step 3: Use the formula for the length of the common tangent
The formula for the length of the common tangent (PQ) between two circles is given by:
\[ PQ = \sqrt{d^2 - (r1 - r2)^2} \]
Where \( (r1 - r2) \) is the absolute difference of the radii.

### Step 4: Calculate \( (r1 - r2) \)
Calculate the difference between the radii:
\[ r1 - r2 = 4 \, \text{cm} - 9 \, \text{cm} = -5 \, \text{cm} \]
We take the absolute value:
\[ |r1 - r2| = 5 \, \text{cm} \]

### Step 5: Substitute the values into the formula
Now substitute the values into the formula for PQ:
\[ PQ = \sqrt{d^2 - (r1 - r2)^2} \]
\[ PQ = \sqrt{13^2 - 5^2} \]
\[ PQ = \sqrt{169 - 25} \]
\[ PQ = \sqrt{144} \]
\[ PQ = 12 \, \text{cm} \]

### Step 6: Calculate the area of the square
The area of a square with side length PQ is given by:
\[ \text{Area} = PQ^2 \]
\[ \text{Area} = 12^2 = 144 \, \text{cm}^2 \]

### Final Answer
Thus, the area of the square is \( 144 \, \text{cm}^2 \).

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