I'd like some help with this rather difficult question:
The tangents at and to the rectangular hyperbola with equation meet on the rectangular hyperbola with equation . Prove that PQ is tangent to the curve with equation
I'm not really sure what I need to do to prove this statement, and I can't seem to find the equation of PQ. Any help would be greatly appreciated.
Equation of tangent:
. So at x = ct, .
Using :
At P: .
At Q: .
Intersection point of these two tangents:
and .
But this intersection point lies on the hyperbola . Sub them in:
.... (A)
Equation of line PQ:
.
Using :
.
Substitute equation (A):
.
Tangent to curve at :
Using the same approach as before (or just make the replacement c --> 2c in the other tangents), you get:
.
So ...... is there a value of that gives , the line PQ?
Compare the tangent and the line:
Gradient (coefficient of x): .
Does this value satisfy ....? Obviously.
So there you go. The line PQ is tangent to , at the point where .