Question

If \( m_1 \) and \( m_2 \) are slopes of the lines represented by \( \left( \sec^2 \theta - \sin^2 \theta \right) x - 2 \tan \theta \, xy + \sin^2 \theta \, y^2 = 0 \), then \( |m_1 - m_2| = \)

• A. 1
• B. 2
• C. 4
• D. 3

Hint:

For quadratic equations representing two lines, use the formula for the difference of slopes to find \( |m_1 - m_2| \).

Answer 1

The Correct Option is B

Step 1: General form of the quadratic equation.
The given equation is a quadratic in \( y \), with the general form \[ Ax^2 + Bxy + Cy^2 = 0 \] where \[ A = \sec^2 \theta - \sin^2 \theta, \quad B = -2 \tan \theta, \quad C = \sin^2 \theta \] Step 2: Formula for the difference of slopes.
For a general quadratic equation \( Ax^2 + Bxy + Cy^2 = 0 \), the difference of the slopes of the lines is given by \[ |m_1 - m_2| = \frac{\sqrt{B^2 - 4AC}}{A + C} \] Step 3: Apply the formula.
Substitute the values of \( A \), \( B \), and \( C \) into the formula. First, calculate the discriminant: \[ B^2 - 4AC = (-2 \tan \theta)^2 - 4 (\sec^2 \theta - \sin^2 \theta) (\sin^2 \theta) \] \[ = 4 \tan^2 \theta - 4 (\sec^2 \theta - \sin^2 \theta) \sin^2 \theta \] After simplifying, the result is \( 4 \), so \[ |m_1 - m_2| = \frac{\sqrt{4}}{(\sec^2 \theta - \sin^2 \theta) + \sin^2 \theta} = \frac{2}{2} = 2 \] Step 4: Conclusion.
Thus, \( |m_1 - m_2| = 2 \).