### Question 1

Consider the graph of the equation:

For various substitutions of real number coefficients a, b and c.

Let us approach this problem by considering some simple graphs first and then building to the question:

What happens when we consider (x - b)(y - a) = 0 with a = b:

Or (x - b) ( y - a ) = 0 with a not equal to b:

I hope that we are getting a pretty clear picture of what is happening. The graph of (x - b)(y - a) = 0 seems to give us a "cross" with a horizontal component equal to the line y = a and vertical component equal to the line x = b.

Let us now introduce a constant, that is let us consider the graph (x - b)(y - a) = k:

Isn't it amazing what a profound impact a small number like 1 can have!!! We now have a rectangular hyperbola with asymptotes y = 3 and x = -2.

What is the impact of varying the constant?

This should not be too surprising!

Let us revisit for a moment rectangular hyperbola (xy = k) that have the axes as their asymptotes. For k greater than 0 the arms are found in the first and third quadrants and the arms intersect the line y = x at the point closest to the origin on each arm - more particularly at the points (sqrt(k),sqrt(k)) and (-sqrt(k), -sqrt(k)). As k increases so will sqrt(k) and the point on the arms closest to the origin moves further and further from the origin. (Clearly the case is analogous for k less than zero).

In the example that we have been dealing with we are simply dealing with this same rectangular hyperbola but have shifted the graph relative to the axes.

If we now expand the brackets in the equation (y - b)(x - a) = k and transpose some of the elements we get:

The equation that we wanted to explore in the first place! This equation is then a rectangular hyperbola with asymptotes at y = b and x = a. The quadrants in which the hyperbola lies will depend on the following:

if c + ab is greater than 0 the hyperbola will lie in the first and third quadrants (determined by the shifted axes), and
if c + ab is less than ) the hyperbola will lie in the second and fourth quadrants (determined by the shifted axes).

Given xy = -4x + 2 y + 4 we predict: asymptotes at y = -4 and x = 2 and a hyperbola that is found in the second and fourth quadrants (4 + (-8) = -4) as determined by the asymptotes (click here to see the graph - but only after you are absolutely happy with the remarks just made).

Try the following by yourself clicking to see the answer only when you have done the necessary thinking:

(i) xy = -3y - 2x - 4
(ii) xy = 3y + 2x - 8
(iii) xy = 3y - 3x + 10
(iv) xy = 3y - 3x + 9

Conclusion

I hope that this has been a useful manner to look at rectangular hyperbola (with shifted axes) - I believe that starting from simple basics helped us to understand the topic with ease and great clarity.

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