Slope Of A Line

The problem asks for the equation of a line that initially passes through the point with an angle of inclination of , after it is rotated about point by in the clockwise direction.

Concept Used:

The solution requires the following concepts from coordinate geometry:

1. Slope of a Line: The slope ( ) of a line that makes an angle with the positive x-axis is given by .
2. Rotation of a Line: When a line with an initial angle of inclination is rotated clockwise by an angle , the new angle of inclination is .
3. Point-Slope Form: The equation of a line passing through a point with a slope is given by the formula .

We will also use the trigonometric identity for the tangent of a difference of angles: .

Step-by-Step Solution:

Step 1: Determine the new angle of inclination of the line after rotation.

The initial angle of the line with the positive x-axis is .

The line is rotated by in the clockwise direction. A clockwise rotation decreases the angle of inclination. Therefore, the new angle is:

Step 2: Calculate the slope ( ) of the line in its new position.

The slope of the new line is given by .

We can find the value of using the angle subtraction formula:

Substituting the standard values and :

To simplify, we rationalize the denominator:

Step 3: Use the point-slope form to find the equation of the new line.

The line passes through the point of rotation and has a slope of .

Final Computation & Result:

Step 4: Check which of the given options matches the derived equation.

The derived equation is .

Let's examine option (1):

Rearranging this equation to solve for :

Factoring out a -1 from both terms:

This matches our derived equation. Therefore, option (2) is the correct answer.

The equation of the line in the new position is .

We are given a limit problem and need to find its value using L'Hopital's Rule. The expression provided is: and it simplifies to after applying the rule.

Step 1: Solve for

The given equation: This equation helps us find the relationship between and .

Step 2: Set up the limit

The limit is given by: We apply L'Hopital's Rule to the expression. First, compute the derivatives of the numerator and denominator.

Step 3: Apply L'Hopital's Rule

Using L'Hopital's Rule: After differentiating both the numerator and the denominator, we proceed with the computation.

Step 4: Compute the limit

Now, substitute into the expression: Finally, simplifying the expression gives:

Final Answer:

The value of the limit is: