Question

The equation $x^2 - 3xy + 2y^2 + 3x - 5y + 2 = 0$ represents a pair of straight lines. If $\theta$ is the angle between them, then the value of $\cos \theta$ is equal to

• A. $\frac{1}{3\sqrt{2}}$
• B. $\frac{3}{\sqrt{10}}$
• C. $\frac{2}{\sqrt{10}}$
• D. $\frac{1}{5\sqrt{2}}$

Hint:

The angle between $ax^2 + 2hxy + by^2 + \dots = 0$ is solely determined by $a, h,$ and $b$.

Answer 1

The Correct Option is B

Step 1: Identify coefficients
$a = 1, h = -3/2, b = 2$.

Step 2: Find $\tan \theta$
$\tan \theta = \frac{2\sqrt{h^2 - ab}}{|a+b|} = \frac{2\sqrt{9/4 - 2}}{|1+2|} = \frac{2\sqrt{1/4}}{3} = \frac{1}{3}$.

Step 3: Convert to $\cos \theta$
If $\tan \theta = 1/3$, then opposite = 1, adjacent = 3.
Hypotenuse = $\sqrt{1^2 + 3^2} = \sqrt{10}$.
$\cos \theta = \frac{3}{\sqrt{10}}$.
Final Answer: (B)