Question

If P and Q are two points whose coordinates are (at² ,2at) and (a/t² , 2a/t) respectively and S is the point (a, 0). Show that `\frac{1}{SP}+\frac{1}{SQ}` is independent of t.

Answer 1

We have,

`SP=sqrt((at^2-a)^2+(2at-0)^2)`

`=sqrt((t^2-1)^2+4t^2)=a(t^2+1)`

`=>SQ=sqrt((a-a/t^2)^2+(0+(2a)/t)^2)`

`=>SQ=sqrt((a^2(1-t^2))/t^4+(4a^2)/t^2)`

`=>SQ=a/t^2sqrt((1-t^2)^2+4t^2)=a/t^2sqrt((1+t^2)^2)`

which is independent of t.

`=\frac{a}{t^{2}}(1 +\t^{2})`

`\therefore\frac{1}{SP}+\frac{1}{SQ}=1/(a(t^2+1))+t^2/(a(t^2+1)`

`\Rightarrow\frac{1}{SP}+\frac{1}{SQ}=(1+t^2)/(a(t^2+1)) = 1/a`