The sporadic complete $12$-arc in $\mathrm{PG}(2,13)$ contains eight points from a conic. In $\mathrm{PG}(2,q)$ with $q>13$ odd, all known complete $k$-arcs sharing exactly $\ha(q+3)$ points with a conic $\mathcal{C}$ have size at most $\frac{1}{2}(q+3)+2$, with only two exceptions, both due to Pellegrino, which are complete $(\frac{1}{2}(q+3)+3)$ arcs, one in $\mathrm{PG}(2,19)$ and another in $\mathrm{PG}(2,43)$. Here, three further exceptions are exhibited, namely a complete $(\frac{1}{2}(q+3)+4)$-arc in $\mathrm{PG}(2,17)$, and two complete $(\frac{1}{2}(q+3)+3)$-arcs, one in $\mathrm{PG}(2,27)$ and another in $\mathrm{PG}(2,59)$. The main result is Theorem 6.1 which shows the existence of a $(\frac{1}{2}(q^r+3)+3)$--arc in $\mathrm{PG}(2,q^r)$ with $r$ odd and $q\equiv 3 \pmod 4$ sharing $\frac{1}{2}(q^r+3)$ points with a conic, whenever $\mathrm{PG}(2,q)$ has a $(\frac{1}{2}(q+3)+3)$-arc sharing $\frac{1}{2}(q+3)$ points with a conic. A survey of results for smaller $q$ obtained with the use of the MAGMA package is also presented

A new proof will be given for the following result originally stated in (Rend. Cl. Sci. Fis. Mat. Natur. (8) 56 (1974) 541): Let $K$ be a complete $k$-arc in $PG(2,q)$, $q$ odd, containing $(q+3)/2$ points from an irreducible conic $\mathcal{C}$ of $PG(2,q)$. If $(q+1)/2$ is a prime, then $K$ contains at most four points outside $\mathcal{C}$. If $q^2\equiv 1\ (\mathrm{mod}\ 16)$, then this number can be at most two

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Motivated by applications to 2-level secret sharing schemes, we investigate k-arcs contained in a (q + 1)-arc of PG(3, q), q even, which have only a small number of focuses on a real axis of . Doing so, we also investigate hyperfocused and sharply focused arcs contained in a translation oval of PG(2, q).