The asymptotes of a hyperbola are y=x+1 and y=3-x. Also, the hyperbola passes through (3,3). Find the distance between the foci of the hyperbola.
The asymptotes of a hyperbola are y=x+1 and y=3-x. Also, the hyperbola passes through (3,3).
Find the distance between the foci of the hyperbola.
Formula hyperbola:
1. asymptotex+1=yx+1−y=02. asymptote3−x=y3−x−y=0hyperbola: (x+1−y)(3−x−y)+c=0P(x=3,y=3)(3+1−3)(3−3−3)+c=0(1)(−3)+c=0c=3hyperbola: (x+1−y)(3−x−y)+3=0(x+1−y)(3−x−y)+3=0((x−y)+1)(3−(x+y))+3=03(x−y)−(x−y)(x+y)+3−(x+y)+3=03(x−y)−(x−y)(x+y)+3−(x+y)+3=03x−3y−(x2−y2)+3−x−y+3=02x−4y−x2+y2+6=0|⋅(−1)x2−2x−y2+4y−6=0x2−2x−y2+4y=6(x−1)2−1−(y−2)2+4=6(x−1)2−(y−2)2=3|:3(x−1)23−(y−2)23=1Formula hyperbola:(x−h)2a2−(x−k)2b2=1a2=3a=√3b2=3b=√3
Focus1:F1(h+ae,k)Focus2:F2(h−ae,k)Foci distance:h+ae−(h−ae)=h+ae−h+ae=2ae|e=√a2+b2a=2a√a2+b2a=2√a2+b2|a2=b2=3=2√3+3=2√6
The distance between the foci of the hyperbola is 2√6