$P$ and $Q$ have coordinates $\left(4, 1\right)$ and $\left(2, 0\right)$. Find: The image $Q\text{'}$ of $Q$  under the reflection in the line $PP\text{'}$, where $P\text{'}$ is the image of $P$ under reflection in the $y-$axis.

$P$ and $Q$ have coordinates $\left(4, 1\right)$ and $\left(2, 0\right)$. If the length of $P\text{'}Q\text{'}$ is $\sqrt{k}$, find the value of $k$.

Points $A\left(5, 1\right)$ is the centre of a circle with radius $13$ units. $AB$ is perpendicular to the chord $PQ$. $B$ is $\left(2, -3\right)$. Calculate the length of  $AB$.

Points $A\left(5, 1\right)$ is the centre of a circle with radius $13$ units. $AB$ is perpendicular to the chord $PQ$. $B$ is $\left(2, -3\right)$. Calculate the length of  $PB$.

Points $A\left(5, 1\right)$ is the centre of a circle with radius $13$ units. $AB$ is perpendicular to the chord $PQ$. $B$ is $\left(2, -3\right)$. Calculate the length of  $PQ$.