HELP! : 2 Parametric Questions!! (1 Viewer)

[kubekoo]

[This is come from 3Units Jones and Couchman Book2 Ex23.5 Q5a and 6c]

Q5a)
If P(2ap,ap^2) and Q(2aq,aq^2) are points on the parabola x=2at, y=at^2, and S is the focus, prove that PS+SQ=a(p^2 +q^2 +2).

and Q6

i know how to do the first 2 parts,

a) A stright line through T(0,-a) cuts the parabola x^2=4ay at P and Q with respective parameters p and q.
Show that the equation of TP is 2py=x(p^2 + 1) - 2ap
b) Prove that for TP to pass through Q, pq=1

and the part i can't do is the following!! ><~

c) Hence prove that 1/SP + 1/SQ = 1/a

><!!! looking forward to heard from any of your guys!~! thanks!!!

 

[withoutaface]

5a) using the focus directrix definition of a parabola, and a diagram, clearly:
SP=ap²+a
SQ=aq²+a
&there4; SP+SQ=a(p²+q²+2)

6c) 1/SP+1/SQ= (SP+SQ)/SP.SQ=a(p²+q²+2)/(ap²+a)(aq²+a)
=a(p²+q²+2)/a²(p²+q²+1+p²q²)
=a(p²+q²+2)/a²(p²+q²+2) [since pq=-1 &there4; p²q²=1]
=1/a

 

[kubekoo]

5a) using the focus directrix definition of a parabola, and a diagram, clearly:
SP=ap²+a
SQ=aq²+a
&there4; SP+SQ=a(p²+q²+2)

um~~~ directrix defintion ?!??! i never remember comes across to that before, can u please explain a bit more !!! thx thx thx!!!!

 

[kubekoo]

um~~~ directrix defintion ?!??! i never remember comes across to that before, can u please explain a bit more !!! thx thx thx!!!!

LOL!!!! thanks ~ I understand now!!!

I am assuming the definition of a parabola is ~ the point is equal distance from a fixed point (focus) and a fixed line (directrix)~
therefore, PS=PM where M is the point P projected onto the directirx!!!~

Thanks you very much!!!!!!!!!!!!!!!!!!