Graphs (1 Viewer)

[coyazayo]

Helloo to everyone reading this thread. Have some questions about graphs I would like to pose.

1. Express hyperobla y=-(3-x)/(2x-1) in the form y=a/x-h +k

2. consider the equation of the hyperbola x=b-cy/y-a. Express the equation in the form of y=a/x-h +k

3. Consider the equations of the straight line and the square root, y=2+ax and y=\sqrt{ 5+x}. Find the set of values of a, when the line intersects the graph of the square root at two distinct points

Thanks in Advance :^)

 

[KingOfActing]

 

[leehuan]

2. consider the equation of the hyperbola x=b-cy/y-a. Express the equation in the form of y=a/x-h +k

Please use brackets. That is very VERY confusing.

Also, note on Q1 whilst I'd vouch for KoA's method anyday, worst comes to worst polynomial long division is fine.
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Without any sneaky tricks, Q3 can be done by equating the two equations, squaring, and then relying on the quadratic discriminant.

 

[KingOfActing]

Please use brackets. That is very VERY confusing.

Also, note on Q1 whilst I'd vouch for KoA's method anyday, worst comes to worst polynomial long division is fine.
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Without any sneaky tricks, Q3 can be done by equating the two equations, squaring, and then relying on the quadratic discriminant.

It can't actually be done that way, the discriminant is always positive. You could probably do some stuff with the conditions ax > -2 and x >= -5, along with the quadratic equation, but imo that seems much harder.

 

[InteGrand]

Yeah, a graphical approach was my first thought for Q3.

 

[leehuan]

Oh that's interesting. Tried it out just now.

 

[Paradoxica]

One could also say by inspection...

 

[KingOfActing]

Yeah the other way to solve 3 is by taking the negative root of the equation once it's squared and applying the two conditions I wrote earlier and solving a quartic (I think) inequality in a.

 

[Paradoxica]

It's not a quartic. It is obvious that by flipping the entire graph of the scenario about the line y=x, we have a (domain restricted) quadratic intersecting a line. Then just consider the case necessary to restrict the branch.

 

[KingOfActing]

It's not a quartic. It is obvious that by flipping the entire graph of the scenario about the line y=x, we have a (domain restricted) quadratic intersecting a line. Then just consider the case necessary to restrict the branch.

true true

still a pain to do tho