The number of lines that can be drawn through the point `(4, -5)` at a distance 12 from the point `(-2, 3)` is

Options

A. 0
B. 1
C. 2
D. infinite

Answer

To solve the problem of finding the number of lines that can be drawn through the point (4, -5) at a distance of 12 from the point (-2, 3), we will follow these steps:

### Step 1: Understand the Problem
We need to find the number of lines that pass through the point (4, -5) and are at a distance of 12 from the point (-2, 3).

### Step 2: Equation of the Line
Let the slope of the line be \( m \). The equation of the line passing through the point (4, -5) can be written as:
\[ y + 5 = m(x - 4) \]
Rearranging this, we get:
\[ y = mx - 4m - 5 \]

### Step 3: Use the Distance Formula
The distance \( d \) from a point \( (x_0, y_0) \) to a line given by \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Here, the line can be rewritten in the form \( Ax + By + C = 0 \):
\[ mx - y - (4m + 5) = 0 \]
Thus, \( A = m \), \( B = -1 \), and \( C = -(4m + 5) \).

### Step 4: Substitute into the Distance Formula
We need to find the distance from the point (-2, 3) to this line, which should equal 12:
\[ d = \frac{|m(-2) + (-1)(3) - (4m + 5)|}{\sqrt{m^2 + 1}} = 12 \]
This simplifies to:
\[ \frac{|-2m - 3 - 4m - 5|}{\sqrt{m^2 + 1}} = 12 \]
Combining terms gives:
\[ \frac{|-6m - 8|}{\sqrt{m^2 + 1}} = 12 \]

### Step 5: Clear the Denominator
Multiplying both sides by \( \sqrt{m^2 + 1} \):
\[ |-6m - 8| = 12\sqrt{m^2 + 1} \]

### Step 6: Consider Two Cases
We will consider two cases based on the absolute value:

**Case 1:**
\[ -6m - 8 = 12\sqrt{m^2 + 1} \]

**Case 2:**
\[ -6m - 8 = -12\sqrt{m^2 + 1} \]

### Step 7: Solve Each Case
1. **For Case 1:**
\[ -6m - 8 = 12\sqrt{m^2 + 1} \]
Square both sides and simplify to find \( m \).

2. **For Case 2:**
\[ -6m - 8 = -12\sqrt{m^2 + 1} \]
Again, square both sides and simplify to find \( m \).

### Step 8: Discriminant Analysis
After solving both cases, we will find the discriminant of the resulting quadratic equations. If the discriminant is less than zero, there are no real solutions, meaning no lines can be drawn.

### Step 9: Conclusion
If both cases yield no real roots, then the number of lines that can be drawn through the point (4, -5) at a distance of 12 from the point (-2, 3) is zero.

### Final Answer
Thus, the number of lines is **0**.

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