Riemannian metrics singular at an isolated point

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Problem 3.2.5 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.2 "Differential Geometry of Surfaces"

Let DD be the disk {(x,y)∈R2:x2+y2<14}\{ (x,y) \in \mathbb {R}^{2} : x^2+y^2 < \tfrac 14\} and let O=(0,0)∈DO = (0,0) \in D. Let (x1,x2)=(r,θ)(x_1,x_2) = (r,\theta ) be the usual polar coordinates and define a metric on D∖{O}D\setminus \{ O\} by

(gij)=(100h(r,θ)), (g_{ij}) = \begin{pmatrix} 1 & 0 \\ 0 & h(r,\theta ) \end{pmatrix},

where h(r,θ)=r2[1−2r2+r4sin⁡2θ]2h(r,\theta ) = r^2[1 - 2r^2 + r^4\sin^2\theta ]^2.

?
(a)

Prove that this metric extends to a smooth metric on DD.

(b)

Show that line segments in DD which pass through the origin (suitably parameterized) are geodesics.

References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.