Let S denote the set of points of intersection of the hyperbola xy = 2 and the graph of y = cube root of . How many lines of slope 1 pass through at least one point of S?
I'm not sure I understand the last bit "How many lines of slope 1 pass through at least one point of S?" .....
You can draw a line of any desired gradient (including m = 1) through any given point (including points in S).
So I would've thought the answer would be equal to the number of points in S .....
Do you mean how many tangents to either xy = 2 or ....? Or, perhaps how many lines passing through two of the points in S ....?
Anyway, you certainly need to know how many points are in S ......
Solve .
Let , say:
.
Therefore and so there are two solutions for x. Therefore there are two points in S.
First calculate the points of intersection:
. Cube both sides, multiply by x³. You'll get an equation:
. Use the substitution t = x³
which will give only 2 x-values. Plug in these x-values into the equation of the hyperbola:
Now use point-slope-formula to calculate the equations of the lines passing through or through .
You'll get in both cases:
To answer your question: There is only one line passing through the whole set of intersection points.
Hello, ihmth
We have: .Let denote the set of points of intersection of the hyperbola
. . and the graph of
How many lines of slope 1 pass through at least one point of ?
Substitute [1] into [2]: .
Quadratic Formula: .
. . Hence: .
Therefore: . has two points: .
Since the slope of is: .
. . there is one line with slope 1 that passes through