Question

If \( \theta \) is the angle between the pair of straight lines \( x^2 - 5xy + 4y^2 + 3x - 4 = 0 \), then \( \tan^2\theta \) is equal to

• A. \( \frac{9}{16} \)
• B. \( \frac{16}{25} \)
• C. \( \frac{9}{25} \)
• D. \( \frac{21}{25} \)
• E. \( \frac{25}{9} \)

Hint:

Always convert \(xy\) coefficient into \(2h\) form before applying formulas.

Answer 1

The Correct Option is C

Concept: For equation: \[ ax^2 + 2hxy + by^2 = 0 \] Angle between lines: \[ \tan\theta = \frac{2\sqrt{h^2 - ab}}{a+b} \]

Step 1: Compare: \[ x^2 - 5xy + 4y^2 \] So: \[ a=1,\quad 2h=-5 \Rightarrow h=-\frac{5}{2},\quad b=4 \]

Step 2: Compute: \[ h^2 - ab = \frac{25}{4} - 4 = \frac{9}{4} \]

Step 3: Substitute: \[ \tan\theta = \frac{2\sqrt{9/4}}{1+4} = \frac{2 \cdot \frac{3}{2}}{5} = \frac{3}{5} \]

Step 4: Square: \[ \tan^2\theta = \frac{9}{25} \]

Step 5: Final answer: \[ \frac{9}{25} \]