If the angle of intersection at a point where the two circles with radii 5 cm and 12 cm intersect is `90^(@)`, then the length (in cm) of their common chord is

Options

A. `(13)/(2)`
B. `(60)/(13)`
C. `(120)/(13)`
D. `(13)/(5)`

Explanation

To find the length of the common chord of two circles with radii 5 cm and 12 cm that intersect at a right angle, we can follow these steps:

### Step 1: Understand the Geometry
Let the centers of the two circles be \( C_1 \) and \( C_2 \), with radii \( r_1 = 12 \, \text{cm} \) and \( r_2 = 5 \, \text{cm} \) respectively. The angle of intersection at the point where the circles intersect is \( 90^\circ \).

### Step 2: Apply the Pythagorean Theorem
Since the angle of intersection is \( 90^\circ \), we can form a right triangle with the centers \( C_1 \) and \( C_2 \) and the intersection point \( M \). The lengths of the segments from the centers to the intersection point are the radii of the circles.

Using the Pythagorean theorem in triangle \( C_1 C_2 M \):
\[ C_1 C_2^2 = C_1 M^2 + C_2 M^2 \]
Substituting the values:
\[ C_1 C_2^2 = r_1^2 + r_2^2 = 12^2 + 5^2 = 144 + 25 = 169 \]
Thus,
\[ C_1 C_2 = \sqrt{169} = 13 \, \text{cm} \]

### Step 3: Find the Length of the Common Chord
Let \( AM \) be the length from the center \( C_1 \) to the midpoint \( M \) of the common chord. The lengths from the centers to the intersection point can be expressed as:
\[ C_1 M^2 = r_1^2 - AM^2 \quad \text{and} \quad C_2 M^2 = r_2^2 - AM^2 \]
Substituting the known values:
\[ C_1 M^2 = 12^2 - AM^2 = 144 - AM^2 \]
\[ C_2 M^2 = 5^2 - AM^2 = 25 - AM^2 \]

### Step 4: Set Up the Equation
From the triangle \( C_1 C_2 M \):
\[ C_1 C_2 = C_1 M + C_2 M \]
Since \( C_1 M + C_2 M = 13 \) cm, we can express \( C_1 M \) and \( C_2 M \) in terms of \( AM \):
\[ C_1 M = \sqrt{144 - AM^2} \quad \text{and} \quad C_2 M = \sqrt{25 - AM^2} \]
Thus,
\[ \sqrt{144 - AM^2} + \sqrt{25 - AM^2} = 13 \]

### Step 5: Solve for \( AM \)
Squaring both sides:
\[ (144 - AM^2) + (25 - AM^2) + 2\sqrt{(144 - AM^2)(25 - AM^2)} = 169 \]
Simplifying:
\[ 169 - 2AM^2 + 2\sqrt{(144 - AM^2)(25 - AM^2)} = 169 \]
This simplifies to:
\[ 2\sqrt{(144 - AM^2)(25 - AM^2)} = 2AM^2 \]
Dividing both sides by 2:
\[ \sqrt{(144 - AM^2)(25 - AM^2)} = AM^2 \]
Squaring again:
\[ (144 - AM^2)(25 - AM^2) = AM^4 \]
Expanding and rearranging gives:
\[ 3600 - 169AM^2 + AM^4 = 0 \]
This is a quadratic in terms of \( AM^2 \).

### Step 6: Solve the Quadratic Equation
Let \( x = AM^2 \):
\[ x^2 - 169x + 3600 = 0 \]
Using the quadratic formula:
\[ x = \frac{169 \pm \sqrt{169^2 - 4 \cdot 3600}}{2} \]
Calculating the discriminant:
\[ 169^2 - 14400 = 28561 - 14400 = 14161 \]
Thus:
\[ x = \frac{169 \pm 119}{2} \]
Calculating the two possible values:
1. \( x = \frac{288}{2} = 144 \)
2. \( x = \frac{50}{2} = 25 \)

### Step 7: Find the Length of the Common Chord
Since \( AM^2 = 25 \):
\[ AM = \sqrt{25} = 5 \, \text{cm} \]
The length of the common chord \( AB \) is \( 2 \times AM = 2 \times 5 = 10 \, \text{cm} \).

### Final Answer
The length of the common chord is \( 10 \, \text{cm} \).