Answer :
The radius to point P in polar coordinates is given by the generic form, [tex]r(ϕ)=d/cos(ϕ−α)[/tex]. The shortest distance between two points is a straight line using polar coordinates.
**a.** Consider a straight line that does not pass through the origin. Let the length of the perpendicular segment between the line and the origin be $d$, and let the polar angle made by the perpendicular segment be $\alpha$. Let $P$ be a point on the line, and let $\phi$ be the polar angle of $P$.
We can see that the radius to point $P$ in polar coordinates is given by the following equation:
[tex]r(\phi) = \frac{d}{\cos(\phi - \alpha)}[/tex]
This equation can be derived by considering the right triangle formed by the line, the perpendicular segment, and the radius to point $P$.
**b.** The shortest distance between two points in Euclidean space is a straight line. This can be shown using the Euler-Lagrange equations.
The Euler-Lagrange equations are a set of differential equations that can be used to find the extremum of a function subject to some constraint. In this case, the function we are trying to extremize is the distance between two points, and the constraint is that the distance must be a straight line.
The Euler-Lagrange equations for this problem can be written as follows:
[tex]\frac{d}{d\phi} \left[ \frac{d}{d\phi} \left[ \frac{1}{2} r^2 (\phi) \right] \right] = 0[/tex]
This equation can be solved to show that the shortest distance between two points is a straight line.
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