State whether the following statement is true or false. Justify your answer.
The point $P(-2, 4)$ lies on a circle of radius $6$ and centre $C(3, 5)$.

(B) False.
$A$ point lies on a circle if and only if the distance between the point and the centre of the circle is equal to the radius of the circle.
Let the centre be $C(3, 5)$ and the point be $P(-2, 4)$. The distance $CP$ is calculated using the distance formula:
$CP = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
$CP = \sqrt{(3 - (-2))^2 + (5 - 4)^2}$
$CP = \sqrt{(3 + 2)^2 + (1)^2}$
$CP = \sqrt{5^2 + 1^2}$
$CP = \sqrt{25 + 1} = \sqrt{26}$
Since $\sqrt{26} \neq 6$,the distance between the point $P$ and the centre $C$ is not equal to the radius of the circle.
Therefore,the point $P(-2, 4)$ does not lie on the circle.