Hi all, this is a really long problem, so please bear with me, parts a - f is done and solution is provided, so it's only g that I need help on, but I figured a - f would help... and once again, any help that you can offer is great!
Here's the question:
The line is a common tangent to two hyperbolas: and , with points of contact and respectively.
(Unfortunately I don't know how to do diagrams on here, so you may have to draw this one out)
a) Show that has equation . - DONE!
b) Show that also has equation . - DONE!
c) Deduce that . - DONE!
d) Write the coordinates of in terms of , , and , and show that . Deduce that there are exactly two such common tangents to the hyperbola. - DONE!
Treating as a quadratic equation in , and consider its discriminant, one can show that there are two common tangents to the hyperbolas.
e) Using the summetry in the graphs to draw in the second common tangent with points of contact on and on . Write the coordinates of and in terms of , , and . - DONE!
f) Show that if is a rhombus, then and deduce that - DONE!
To show , simply solve
The right hand side is strictly negative (as and ), therefore the left side must also be negative.
g) Show that if is a square, then and deduce that . What is the relationship between the two hyperbolas if is a square?
NEED HELP: I can't show the first part... but the deduction I can do once the first part is done... and then I haven't yet attempted the relationship (but I'm sure I'll have no trouble doing that)
THANKS A LOT GUYS, IT'LL REALLY HELP ME OUT IF YOU COULD GIVE ME A HAND WITH PART g)