Please help to find the equation for this hyperbola problem

[minivan09ia]

Homework Statement

I would really appreciate if someone can help me understand how my teacher came up with the answer for this find my equation hyperbola problem. I have enclosed a copy of the problem and the solution provided by my teacher. Thanks.

Homework Equations

The Attempt at a Solution

 

[Simon Bridge]

Welcome to PF;
Looks like you are provided with a graph of a hyperbola with some points on it, and a labelled assymptote.
(One of the points is a hole.)

In principle you should be able to just read five points off the graph to solve the conic section equation - but you are best to use points that are given to you and so avoid making an error reading the graph.

It is faster to use reason and the clues provided ... like if you know how hyperbolas factor in general, then you can pretty much plug in the values for the roots and the assymptotes and limits... whose values you can read off the graph.

I cannot tell from the pics how your teacher did it.
Have you had a go reasoning it out yourself?

The underlined (x-3) appears in the solution numerator and denominator ... there is a hole at x=3, but the inclusion in the answer cancels out.

 

[minivan09ia]

Thank you for the quick reply.

I managed to solve the problem by using the reasons and clues like you suggested. By using the zeros, vertical asymptote, and hole, I found the solution.

However, I am still wondering how the equation would change when the hyperbola opens east-west instead of north-south. Any ideas?

 

[Simon Bridge]

You mean if you rotate it 90deg about the origin?
Why not do the transform and see?