Question

For `f(t)= (1/(1-t),-1/(1+t^2))` what is the distance between `f(2)` and `f(5)`?

Answer 1

`d=(3sqrt4421)/(260)approx0.767199`

Explanation:

Find the points first:

`f(2)=(1/(1-2),-1/(1+2^2))=(-1,-1/5)`

`f(5)=(1/(1-5),-1/(1+5^2))=(-1/4,-1/26)`

The distance between the points `(x_1,y_1)` and `(x_2,y_2)` is calculated through `d=sqrt((y_2-y_1)^2+(x_2-x_1)^2)`.

`d=sqrt((-1/26- (-1/5))^2+(-1/4-(-1))^2)`

`d=sqrt((21/130)^2+(3/4)^2)`

You can go ahead and make this a decimal answer, but you can also do some nice simplification if you feel up to it.

`d=sqrt(((3*7)/(2*5*13))^2+(3/2^2)^2)`

`d=sqrt((3^2*7^2)/(2^2*5^2*13^2)+3^2/2^4)`

`d=sqrt((2^2*3^2*7^2)/(2^4*5^2*13^2)+(3^2*5^2*13^2)/(2^4*5^2*13^2))`

`d=sqrt((3^2(2^2*7^2+5^2*13^2))/(2^4*5^2*13^2))`

`d=3/(2^2*5*13)sqrt((2*7)^2+(5*13)^2)`

`d=(3sqrt4421)/(260)approx0.767199`