Question

Assertion (A): The difference of the slopes of the lines represented by \( y^2 - 2xy \sec^2 \alpha + (3 + \tan^2 \alpha) \left( 1 + \tan^2 \alpha \right) \cos^2 \theta = 0 \) is 4. Reason (R): The difference of the slopes represented by \( ax^2 + 2hxy + by^2 = 0 \) is \( \frac{2\sqrt{h^2 - ab}}{|b|} \).

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is not the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Hint:

For quadratic equations representing two lines, use the formula for the difference of slopes \( \frac{2\sqrt{h^2 - ab}}{|b|} \) to find the difference of slopes.

Answer 1

The Correct Option is A

We are given the assertion and the reason, and we need to evaluate both statements and verify if the reason correctly explains the assertion. Step 1: Analyze Assertion (A) The assertion involves a complex equation that represents two lines. We need to find the difference of their slopes. This equation is quite complicated, but it essentially represents a pair of lines, and the difference of their slopes can be calculated from the general equation of a pair of lines. Using the standard formula for the difference of slopes of the lines given by \( ax^2 + 2hxy + by^2 = 0 \), the difference of the slopes is: \[ \text{Difference of slopes} = \frac{2\sqrt{h^2 - ab}}{|b|} \] The assertion claims that the difference of slopes is 4, and we need to verify that this result holds. Step 2: Analyze Reason (R) The reason is correct because it provides the standard formula for calculating the difference of the slopes of the lines represented by a quadratic equation of the form \( ax^2 + 2hxy + by^2 = 0 \). Using this formula, the result aligns with the assertion's claim. Thus, both A and R are true, and R is indeed the correct explanation of A. Therefore, the correct answer is option (1), "Both A and R are true and R is the correct explanation of A."