To solve the problem, we need to find the total number of different chords of a circle that pass through a point \( P \) and have integral lengths. Let's break down the solution step by step.
### Step 1: Understand the given information
We have a circle with:
- Radius \( r = 15 \) units
- Distance from the center of the circle to point \( P \) is \( d = 9 \) units
### Step 2: Determine the maximum length of the chord
The maximum length of a chord in a circle is equal to the diameter of the circle. The diameter \( D \) can be calculated as:
\[
D = 2 \times r = 2 \times 15 = 30 \text{ units}
\]
### Step 3: Use the Pythagorean theorem to find the minimum length of the chord
To find the minimum length of the chord passing through point \( P \), we can use the Pythagorean theorem. The half-length of the chord \( L/2 \) can be calculated as:
\[
\left( \frac{L}{2} \right)^2 + d^2 = r^2
\]
Substituting the known values:
\[
\left( \frac{L}{2} \right)^2 + 9^2 = 15^2
\]
This simplifies to:
\[
\left( \frac{L}{2} \right)^2 + 81 = 225
\]
\[
\left( \frac{L}{2} \right)^2 = 225 - 81 = 144
\]
Taking the square root:
\[
\frac{L}{2} = \sqrt{144} = 12
\]
Thus, the full length of the chord \( L \) is:
\[
L = 2 \times 12 = 24 \text{ units}
\]
### Step 4: Identify the range of integral lengths
Now, we have:
- Minimum chord length \( = 24 \) units
- Maximum chord length \( = 30 \) units
The integral lengths of the chords that can be formed between these two lengths are:
\[
24, 25, 26, 27, 28, 29, 30
\]
### Step 5: Count the number of different integral lengths
The integral lengths are:
- 24
- 25
- 26
- 27
- 28
- 29
- 30
This gives us a total of 7 different integral lengths.
### Step 6: Consider the symmetry of chords
For each integral length \( L \) (except the maximum and minimum), there are two chords (one on each side of point \( P \)). Therefore, we count:
- For lengths 25, 26, 27, 28, and 29, there are 2 chords each.
- For lengths 24 and 30, there is 1 chord each.
Thus, the total number of chords is:
\[
1 (for \, 24) + 2 (for \, 25) + 2 (for \, 26) + 2 (for \, 27) + 2 (for \, 28) + 2 (for \, 29) + 1 (for \, 30) = 1 + 2 + 2 + 2 + 2 + 2 + 1 = 12
\]
### Final Answer
The total number of different chords of the circle passing through point \( P \) and having integral lengths is **12**.
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