Find the vertices, eccentricity, foci and asymptotes of the hyperbola (x2/8) − (y2 /4) = 1. Also trace it. Under what conditions on λ the line x+λy = 2 will be tangent to this hyperbola? Explain geometrically.
1
Expert's answer
2021-06-12T05:23:24-0400
We compare this equation with a2x2−b2y2=1.
Eccentricity is e=1+a2b2=23
The center is C=(0,0)
The vertices are V′=(−a,0)=(−22,0) and V=(a,0)=(22,0)
To find the foci, we need the distance from the center to the foci c2=a2+b2=12,c=±23
The foci are F′=(−c,0)=(23,0) and F=(c,0)=(−23,0)
The asymptotes are 8x2−4y2=0,y=±21x
We compare equation of tangent to hyperbola x0⋅a2x−y0⋅b2y=1 with x+λy=2
We have x−b2a2x0y0y=x0a2,x−2x0y0y=x08 and x0=4, so y0=±2 and λ=−2y0=∓1
Comments
Leave a comment