Question

If the acute angle between the lines \[ x^2-4xy+y^2=0 \] is \(\tan^{-1}(k)\), then \(k=\)

• A. \(\dfrac{1}{\sqrt{3}}\)
• B. \(\sqrt{3}\)
• C. \(\dfrac{1}{6}\)
• D. \(\dfrac{1}{3}\)

Hint:

For pair of lines through origin, remember the direct formula for the angle using coefficients.

Answer 1

The Correct Option is B

Step 1: Compare with the standard form.
For a pair of straight lines through the origin: \[ ax^2+2hxy+by^2=0 \] Here \(a=1,\;2h=-4\Rightarrow h=-2,\;b=1\).
Step 2: Use the angle formula.
The angle \(\theta\) between the lines is given by \[ \tan\theta=\frac{2\sqrt{h^2-ab}}{a+b} \]
Step 3: Substitute values.
\[ \tan\theta=\frac{2\sqrt{(-2)^2-(1)(1)}}{1+1} =\frac{2\sqrt{3}}{2}=\sqrt{3} \]
Step 4: Identify \(k\).
Since \(\theta=\tan^{-1}(k)\), we get \[ k=\sqrt{3} \]