Find the Properties x^2-y^2=1

Problem

x2−y2=1

Solution

1. Identify the type of conic section by comparing the equation to the standard form (x2)/(a2)−(y2)/(b2)=1 This is a horizontal hyperbola.

2. Determine the values of a and b from the denominators. Since a2=1 and b2=1 we find a=1 and b=1

3. Calculate the distance to the foci using the relation c2=a2+b2 Here, c2=1+1=2 so c=√(,2)

4. Locate the center, which is at the origin (0,0) because there are no shifts in x or y

5. Find the vertices by moving a units from the center along the transverse axis (x-axis). The vertices are (±1,0)

6. Find the foci by moving c units from the center along the transverse axis. The foci are (±√(,2),0)

7. Determine the equations of the asymptotes using the formula y=±b/a*x Substituting the values gives y=±x

8. Calculate the eccentricity using e=c/a Here, e=√(,2)/1=√(,2)

Final Answer

Center: *(0,0), Vertices: *(±1,0), Foci: *(±√(,2),0), Asymptotes: *y=±x, Eccentricity: √(,2)

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