Question

Find the directrix of hyperbola , x2 - y2 = 1. 
1. x = \(\pm \frac{1}{\sqrt{2}}\)
2. x = \(\pm \frac{1}{2}\)
3. x = ±1 
4. y = 0

Answer 1

Correct Answer - Option 1 : x = \(\pm \frac{1}{\sqrt{2}}\)

Concept: ​

Equation of hyperbola , \(\rm \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}= 1\) 

Eccentricity, e = \(\rm\sqrt{1+\frac{b^{2}}{a^{2}}}\)  

Directrix, x  = \(\rm\pm \frac{a}{e}\) 

 

Equation of hyperbola , \(\rm- \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}= 1\)  

Eccentricity, e = \(\rm\sqrt{1+\frac{a^{2}}{b^{2}}}\) 

Directrix, y = \(\rm\pm \frac{b}{e}\)  

 

Calculation: 

Given hyperbolic equation ,  x2 - y2 = 1 

⇒ \(\rm \frac{x^{2}}{1}-\frac{y^{2}}{1}= 1\) 

On comparing with standard equation,  a = 1 and b = 1 .

We know that eccentricity, e = \(\rm\sqrt{1+\frac{b^{2}}{a^{2}}}\) 

⇒ e = \(\rm\sqrt{1+\frac{1^{2}}{1^{2}}}\) 

⇒ e = \(\sqrt{2}\)

As we know that , Directrix , x = \(\rm\pm \frac{a}{e}\) 

⇒ Directrix, x = \(\pm \frac{1}{\sqrt{2}}\) 

The correct option is 1 .