Write down the equation for the paths with the following image sets

Question

i)The straight line connecting the points $1+i$ and $3-i$ where $1+i$ is the initial point and $3-i$ is the final point.

ii)The circular arc with the initial point $3+i$, final point $1+3i$ and centre $1+i$

Now I found these formulas in my notes that gave the equations for straight lines and circular arc and by putting the points into these I got this

i) $\gamma(t)=(1-t)(1+i)+t(3-i)$

ii) $\gamma(t)=1+i+2e^{it}$

However I have looked at some other examples and I am unsure as to whether this is correct.

Oh also my backslash button is broken so I someone could change the forward slashes to backslashes that would be greatly appreciated.

Answer 1

Your parametrizations are mostly correct (but there are other possible answers), but you should include the interval for $t$.

i) fine, $t$ should vary from $0$ to $1$

ii) also fine, but here $t$ should vary from $0$ to $\dfrac{\pi}{2}$.