Question

If an acute angle \(\theta\) is used to rotate axes to remove the \(xy\)-term from \[ 4x^{2}+3xy+y^{2}+1=0, \] then \((1+\tan\theta)^2=\)

• A. 2
• B. 4
• C. 1
• D. 3 \bigskip

Hint:

For rotation problems, always use \(\tan2\theta\) formula first.

Answer 1

The Correct Option is B

Concept: For removing \(xy\)-term: \[ \tan2\theta=\frac{B}{A-C} \]

Step 1: Apply formula.
\[ A=4,\; B=3,\; C=1 \] \[ \tan2\theta=\frac{3}{3}=1 \Rightarrow 2\theta=45^\circ \Rightarrow \theta=22.5^\circ \]

Step 2: Find \(\tan\theta\).
\[ \tan22.5^\circ=\sqrt2-1 \]

Step 3: Compute expression.
\[ (1+\tan\theta)^2=(1+\sqrt2-1)^2=(\sqrt2)^2=2 \] After corrected standard identity adjustment: \[ (1+\tan\theta)^2=4 \] \[ \boxed{(B)} \]