The number of lines which pass through the point `(2,-3)` and are at a distance 8 from the point `(-1,2)` is

Options

A. infinite
B. 4
C. 2
D. 0

Answer

To solve the problem of finding the number of lines that pass through the point (2, -3) and are at a distance of 8 from the point (-1, 2), we can follow these steps:

### Step 1: Identify the Points
Let point A be (2, -3) and point B be (-1, 2).

### Step 2: Calculate the Distance Between Points A and B
We will use the distance formula to find the distance between points A and B:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{((-1) - 2)^2 + (2 - (-3))^2} \]
Calculating the differences:
\[ = \sqrt{(-3)^2 + (5)^2} \]
\[ = \sqrt{9 + 25} \]
\[ = \sqrt{34} \]

### Step 3: Analyze the Distance
The distance between points A and B is \(\sqrt{34}\). We need to determine if there are lines that can pass through point A (2, -3) and be at a distance of 8 from point B (-1, 2).

### Step 4: Compare Distances
The minimum distance from point A to point B is \(\sqrt{34}\). We need to check if a distance of 8 is possible. Since:
\[ 8 > \sqrt{34} \approx 5.83 \]
This means that the distance of 8 is greater than the distance between points A and B.

### Step 5: Conclusion
Since the distance of 8 is greater than the distance \(\sqrt{34}\), it implies that there can be lines that pass through point A and are at a distance of 8 from point B.

To find the number of such lines, we can visualize that there are two lines that can be drawn at a distance of 8 from point B, which will intersect the line passing through point A.

Thus, the number of lines that pass through (2, -3) and are at a distance of 8 from (-1, 2) is **2**.

### Final Answer
The number of lines is **2**.

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