Rectangular Hyperbola (1 Viewer)

[zeek]

There are two question's that i can't get for the rectangular hyperbola in Cambridge, exercise 3.4 Q11 and 12.

11. P and Q are variable points on the rectangular hyperbola xy=9. The tangents at P and Q meet at R. If PQ passes through the point (6,2), find the equation of the locus of R.

12. P and Q are variable points on the rectangular hyperbola xy=c². The tangents at P and Q meet at R. If PQ passes through the point (a,0), find the equation of the locus of R.

For both of the i end up with pq remaining and i can't get rid of it

 

[Wackedupwacko]

first question :

eqn of PQ : x+pqy= 3(p+q)
sub in (6,2) 6+2pq=3(p+q)
rearrange into 2pq=3(p+q)-6

R has the coords

y = 6 / (p+q)

x =6pq / (p+q)

=> x= 3(3(p+q) - 6 ) / (p+q)
=> x = 9- 18/(p+q)
=> x = 9-3y

2nd Question:
eqn of PQ : x+pqy= c(p+q)
sub (a,0) : a= c(p+q)
p+q = a/c

R has coords
y = 2c/ (p+q)
x = 2cpq / (p+q)

=> y = 2c/ (a/c)
=> y = 2c²/a

and thats ur locus a straight line

 

[STx]

which did u sub in which in q1)

 

[Wackedupwacko]

you sub the point (6,2) into the eqn of PQ

so it becomes 2pq = 3(p+q) - 6

then in x = 6pq/ (p+q) replace 2pq with above to get

3(3(p+q)-6) / (p+q)
= 9-3y