Question #277295

Show that if B>0, then the graph of x²+Bxy=F, is a hyperbola if F≠0, and two intersecting lines if F=0.

1
Expert's answer
2021-12-09T13:52:23-0500

A conic in the two-dimensional plane is the solution set of an equation of the form


ax2+bxy+cy2+dx+ey+f=0ax^2 + bxy + cy^2 + dx + ey +f = 0

for real parameters a,b,c,d,ea,b,c,d,e and ff, where at least one of a,ba,b and cc is not zero.

If b24ac>0,b^2-4ac>0, then the graph is a hyperbola (or two intersecting lines).

We have


x2+Bxy=Fx^2+Bxy=F

a=1,b=B,c=0,f=Fa=1, b=B, c=0, f=-F

b24ac=B24(1)(0)=B2>0,B0b^2-4ac=B^2-4(1)(0)=B^2>0, B\not=0

Therefore the graph of x2+Bxy=F,x^2+Bxy=F, is a hyperbola or two intersecting lines., if B0.B\not=0.

If F0,F\not=0, we have a hyperbola


y=xB+F/Bxy=-\dfrac{x}{B}+\dfrac{F/B}{x}

If F=0,x2+Bxy=0,F=0,x^2+Bxy=0, we have two intersecting lines

x=0 or y=1Bxx=0\ or\ y=-\dfrac{1}{B}x

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