Question

One equation of common tangent to ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 2$ is:

• A. 2y = $\sqrt{3}$bx + ab
• B. y = 2$\sqrt{3}\frac{b}{a}$x + 2b
• C. No common tangent
• D. ay = $\sqrt{3}$bx + 2ab

Answer 1

Answer: (D)

ay = $\sqrt{3}$bx + 2ab

Answer 2

y = mx + $\sqrt{a^{2}m^{2} + b^{2}}$

y = mx + $\sqrt{2a^{2}m^{2} - 2b^{2}}$

a²m² + b² = 2a²m² - 2b²

a²m² = 3b² ⇒ m = $\pm \frac{\sqrt{3}b}{a}$

$y = \frac{\sqrt{3}b}{a}x + 2b$.