The number of rational points on the line joining `(sqrt(5), 3)` and `(3, sqrt(3))` is

Options

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

Explanation

To find the number of rational points on the line joining the points \((\sqrt{5}, 3)\) and \((3, \sqrt{3})\), we will follow these steps:

### Step 1: Find the slope of the line
The slope \(m\) of the line joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, we have the points \((x_1, y_1) = (\sqrt{5}, 3)\) and \((x_2, y_2) = (3, \sqrt{3})\).

Substituting the values:
\[ m = \frac{\sqrt{3} - 3}{3 - \sqrt{5}} \]

### Step 2: Analyze the numerator and denominator
Both the numerator \((\sqrt{3} - 3)\) and the denominator \((3 - \sqrt{5})\) are irrational numbers.

### Step 3: Determine the rational points
Since both the slope \(m\) is irrational, the line connecting the two points will not contain any rational points. A line can only have rational points if the slope is rational.

### Conclusion
Thus, the number of rational points on the line joining the points \((\sqrt{5}, 3)\) and \((3, \sqrt{3})\) is:
\[ \text{Number of rational points} = 0 \]

### Final Answer
The answer is \(0\).

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