State whether the following statement is true or false. Justify your answer.
$A$ circle has its centre at the origin and a point $P(5,0)$ lies on it. The point $Q(6,8)$ lies outside the circle.

(TRUE) True.
Given that the circle has its centre at the origin $O(0,0)$ and the point $P(5,0)$ lies on the circle.
The radius $r$ of the circle is the distance between the centre $O(0,0)$ and the point $P(5,0)$ on the circle.
Using the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$:
$r = OP = \sqrt{(5 - 0)^2 + (0 - 0)^2} = \sqrt{5^2 + 0^2} = \sqrt{25} = 5$.
Now,we calculate the distance of point $Q(6,8)$ from the centre $O(0,0)$:
$OQ = \sqrt{(6 - 0)^2 + (8 - 0)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$.
We know that if the distance of a point from the centre is greater than the radius,the point lies outside the circle.
Since $OQ = 10$ and $r = 5$,we have $OQ > r$.
Therefore,the point $Q(6,8)$ lies outside the circle. Hence,the statement is true.