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."

Answer 2

Step 1: Understand the form of the equation
The given equation is:
\( y^2 - 2xy \sec^2 \alpha + (3 + \tan^2 \alpha)(1 + \tan^2 \alpha)\cos^2 \theta = 0 \)
We write this in the general second-degree form:
\( ax^2 + 2hxy + by^2 = 0 \)
Here,
\( a = (3 + \tan^2 \alpha)(1 + \tan^2 \alpha)\cos^2 \theta \),
\( h = -\sec^2 \alpha \),
\( b = 1 \)

Step 2: Use formula for difference of slopes
For a homogeneous equation of the form \( ax^2 + 2hxy + by^2 = 0 \), the lines intersect at the origin, and the difference of slopes is:
\[ \Delta m = \frac{2\sqrt{h^2 - ab}}{|b|} \]
This is exactly what is given in the Reason (R) → ✅

Step 3: Apply this to the given equation
Let us compute \( h^2 - ab \):
\[ h = -\sec^2 \alpha \Rightarrow h^2 = \sec^4 \alpha \]
\[ a = (3 + \tan^2 \alpha)(1 + \tan^2 \alpha)\cos^2 \theta \]
\[ ab = a \cdot 1 = a \]
So, \[ \Delta m = \frac{2\sqrt{\sec^4 \alpha - (3 + \tan^2 \alpha)(1 + \tan^2 \alpha)\cos^2 \theta}}{1} \]
We are told this value equals 4. Therefore, the assertion is that the value of this expression equals 4 → ✅

Step 4: Check logical connection
The reason directly provides the formula used to compute the difference of slopes in the assertion. Since the formula is correctly applied and results in the value 4, the Reason (R) is the correct explanation of Assertion (A).

Final Answer:
Both A and R are true and R is the correct explanation of A.