Conics Q (1 Viewer)

[ordnaiv]

hey guys, sorry to be a pest but my mind is just goin blank on this cambridge exercise 3.2 question:

P(acos@, bsin@) and Q (acos#, bsin#) lie on an ellipse:

If PQ subtends a right angle at (a,o) show that tan(@/2)tan(#/2) = - b^2/a^2

 

[haque]

let A(a,0) then m(PA)=bsin@/a(cos@-1) and m(QA)=bsin#/a(cos#-1) and cos@-1=1-2sin^2@/2-1=-2sin^2@/2 and using this for both "m"s we get
m(PA)*m(QA)=-1
b^2sin@sin#/a^2(-2sin^2@/2)(-2sin^2#/2)=-1
4sin@/2 cos@/2 sin#/2 cos#/2/(4sin^2@/2sin^2#/2=-a^2/b^2
cot@/2 cot#/2=-a^2/b^2
and so tan@/2 tan#/2=-b^2/a^2