$$Statement-1$$: If the angle between two asymptotes of a hyperbola $$x2a2$$−$$y2b2=1$$ is π$$3$$ its eccentricity is $$23$$.Statement $$2$$: Angles between the asymptotes of the hyperbola $$x2a2$$−$$y2b2=1$$ are $$2tan$$−$$1$$⁡ba or π−$$2tan$$−$$1$$⁡ba.

Question

$$Statement-1$$: If the angle between two asymptotes  of a hyperbola  $$x2a2$$−$$y2b2=1$$ is π$$3$$ its eccentricity is $$23$$.Statement $$2$$: Angles between the asymptotes of the hyperbola $$x2a2$$−$$y2b2=1$$ are $$2tan$$−$$1$$⁡ba or  π−$$2tan$$−$$1$$⁡ba.

Infinity Learn · Accepted Answer

Equation of the asymptotes is xa±$$yb=0And$$ if θ is an angle between the asymptotes $$tan$$ θ $$=2ba1+b2a2=2tan$$⁡ϕ$$1+tan2$$⁡ϕ where ϕ$$=$$$$tan$$−$$1$$⁡$$ba=$$$$tan$$⁡$$2$$ϕ⇒θ$$=2$$ϕ$$=2tan$$−$$1$$⁡ba and the $$statement-1$$ is true. Using in $$statement-1$$. If π$$3=2tan$$−$$1$$⁡ba⇒$$tan$$⁡π$$6=ba$$⇒$$3b2=a2$$⇒$$3e2$$−$$1=1$$⇒$$e=23$$ If π$$3=$$π−$$2tan$$−$$1$$⁡ba⇒$$ba=$$$$tan$$⁡π$$3=3$$⇒$$b2=3a2$$⇒$$e2$$−$$1=3$$⇒$$e=2Showing$$ that $$statement-1$$ is false

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

Statement-1: If the angle between two asymptotes of a hyperbola x2a2−y2b2=1 is π3 its eccentricity is 23.Statement 2: Angles between the asymptotes of the hyperbola x2a2−y2b2=1 are 2tan−1⁡ba or π−2tan−1⁡ba.

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material