The function represents a hyperbola. Use the asymptotes of the hyperbola (1.3.4.22) in order to calculate the angle of rotation necessary for transforming it to canonical form!
Do the rotation and calculate the corresponding values of a and b of its canonical form:
We have to add to the asymplotes also the lines x=c corresponding to the points:
(1.3.4.26)
Parts 1-3
Solution of question 1
From (1.3.4.21) one finds the slope of an asymptote:
The same result is obtained with
.
The intercept is obtained from (1.3.4.17):
and finally this asymptote is
An additional asymptote corresponds to the divergence of the function to infinity (1.3.4.26) and is:
Parts 4-6
Solution of question 2
From the solution of question 1, we have two angular segments of the polar angle
The function should be contained in one of them.
In order to find that, we can take a positive value of x, calculate y and look in which segment the point is situated.
For example x=1 yields y=0, and therefore the right-hand segment from part 4 is the required one.
We have to rotate the curve so that the mid angle of this interval:
becomes
after rotation.
In other words we need a rotation of
Parts 7-9
Solution of question 3
From the substitution of (1.2.5.35) in the function
and by denoting
we have
But since
the mixed terms with x'y' vanish and we are left with
From part 8 and by using the numerical values of part 7 one obtains
Score
Correct solutions give credit according to the questions:
Question 1 is worth 2 points.
Question 2 is worth 5 points.
Question 3 is worth 3 point.
If the sum of the credits by parts is not an integer, it should be rounded up to the nearest integer.
Parts 1,2 are worth half a point each.
Parts 3,5,7,8,9 are worth one point each.
Parts 4,6 are worth 2 points each.