The number of points on X-axis which are at a distance c units `(c lt 3)` from (2, 3) is

Options

A. 1
B. 2
C. 0
D. 3

Answer

To solve the problem of finding the number of points on the X-axis that are at a distance \( c \) units (where \( c < 3 \)) from the point \( (2, 3) \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Coordinates on the X-axis**:
The points on the X-axis can be represented as \( (x, 0) \), where \( x \) is any real number.

**Hint**: Remember that any point on the X-axis has a y-coordinate of 0.

2. **Use the Distance Formula**:
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, we want the distance from the point \( (2, 3) \) to the point \( (x, 0) \) to equal \( c \). Therefore, we set up the equation:
\[ \sqrt{(x - 2)^2 + (0 - 3)^2} = c \]

**Hint**: Substitute the coordinates into the distance formula correctly.

3. **Simplify the Equation**:
Squaring both sides to eliminate the square root gives:
\[ (x - 2)^2 + 9 = c^2 \]
Rearranging this, we have:
\[ (x - 2)^2 = c^2 - 9 \]

**Hint**: Remember to isolate the squared term correctly.

4. **Analyze the Right Side**:
Since \( c < 3 \), it follows that \( c^2 < 9 \). This means that \( c^2 - 9 < 0 \). Therefore, we have:
\[ (x - 2)^2 = c^2 - 9 < 0 \]

**Hint**: Recognize that a square of a real number cannot be negative.

5. **Conclusion**:
Since \( (x - 2)^2 \) is always non-negative (i.e., \( \geq 0 \)), and we have derived that it must be less than zero, this leads to a contradiction. Hence, there are no real values of \( x \) that satisfy the equation.

Therefore, the number of points on the X-axis that are at a distance \( c \) units from the point \( (2, 3) \) is **zero**.

**Final Answer**: The number of points is \( 0 \).