The number of points on the line `3x +4y = 5`, which are at a distance of `sec^(2)theta +2 cossec^(2) theta, theta in R`, from the point (1,3) is (a) 1 (b) 2 (c) 3 (d) infinite

Options

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

Explanation

To solve the problem, we need to determine the number of points on the line given by the equation \(3x + 4y = 5\) that are at a distance of \(d = \sec^2 \theta + 2 \csc^2 \theta\) from the point \((1, 3)\).

### Step 1: Find the distance from the point to the line

The formula for the distance \(D\) from a point \((x_0, y_0)\) to a line \(Ax + By + C = 0\) is given by:

\[ D = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

For the line \(3x + 4y = 5\), we can rewrite it in the form \(Ax + By + C = 0\):

\[ 3x + 4y - 5 = 0 \]

Here, \(A = 3\), \(B = 4\), and \(C = -5\). The point \((x_0, y_0) = (1, 3)\).

Now, substituting these values into the distance formula:

\[ D = \frac{|3(1) + 4(3) - 5|}{\sqrt{3^2 + 4^2}} = \frac{|3 + 12 - 5|}{\sqrt{9 + 16}} = \frac{|10|}{5} = 2 \]

### Step 2: Analyze the expression for distance

We are given that the distance from the point \((1, 3)\) to the points on the line is equal to \(d = \sec^2 \theta + 2 \csc^2 \theta\).

### Step 3: Determine the minimum value of \(d\)

To find the minimum value of \(d\), we can express \(\sec^2 \theta\) and \(\csc^2 \theta\) in terms of \(\tan^2 \theta\) and \(\cot^2 \theta\):

\[ d = \sec^2 \theta + 2 \csc^2 \theta = 1 + \tan^2 \theta + 2(1 + \cot^2 \theta) = 3 + \tan^2 \theta + 2 \cot^2 \theta \]

Using the AM-GM inequality, we know that:

\[ \tan^2 \theta + 2 \cot^2 \theta \geq 3\sqrt[3]{\tan^2 \theta \cdot \tan^2 \theta \cdot \cot^2 \theta} = 3 \]

Thus, the minimum value of \(d\) is:

\[ d \geq 3 + 3 = 6 \]

### Step 4: Compare the distances

From Step 1, we found that the distance from the point \((1, 3)\) to the line is \(2\). Since the minimum value of \(d\) is \(6\) and the distance from the point to the line is \(2\), we can conclude that:

\[ d \geq 6 > 2 \]

### Step 5: Conclusion

Since the distance \(d\) cannot equal \(2\) (the distance from the point to the line), there are no points on the line \(3x + 4y = 5\) that are at a distance of \(d\) from the point \((1, 3)\).

Thus, the answer is:

**(a) 0 points.**