Question

The angle between lines joining the origin to the points of intersection of the line $\sqrt{3}$x + y = 2 and the curve y² - x² = 4 is:

• A. $\tan^{- 1}\left( \frac{2}{\sqrt{3}} \right)$
• B. $\frac{\pi}{6}$
• C. $\tan^{- 1}\left( \frac{\sqrt{3}}{2} \right)$
• D. $\frac{\pi}{2}$

Answer 1

Answer: (C)

$\tan^{- 1}\left( \frac{\sqrt{3}}{2} \right)$

Answer 2

On homogenizing y² - x² = 4 with help of the line $\sqrt{3}$x + y = 2

y² - x² = 4$\frac{(\sqrt{3}x + y)^{2}}{4}$

⇒ y² - x² = 3x² + y² + 2$\sqrt{3}$ xy

⇒ 4x² + 2$\sqrt{3}$ xy = 0

tanθ = $\frac{2\sqrt{h^{2} - ab}}{a + b}$

$\tan\theta = \frac{2\sqrt{3 - 0}}{4}$

θ = tan^-1 $\left( \frac{\sqrt{3}}{2} \right)$.