Question

In the figure $C(3,0)$ is the centre of the circle and its radius is $5$ units.
(a) Find the co-ordinates of the points $A,B,P$ and $Q$.
(b) Find the co-ordinates of another point on the circle.
(c) Check whether the point $(0,5)$ is inside the circle.

Answer 1

(a) Centre of the circle is $C(3,0)$
$\Rightarrow$ Co-ordinates of $B\equiv (3,+5,0)\equiv (8,0)$
Subtracting $5$ units along the negative $X$ direction from the centre, the co-ordinates of $A$ are $(3,-5,0)\equiv (-2,0)$
$△COQ$ is a right triangle
$\Rightarrow {OC}^2+{OQ}^2={CQ}^2$
$\Rightarrow 3^2+{OQ}^2=5^2$
$\Rightarrow {OQ}^2=25-9=16$
$\Rightarrow$ $OQ=4$ units
$\therefore$ Co-ordinates of $Q$ are $Q(0,-4)$
$\therefore$ Co-ordinates of $P$ are $(0,4)$
(b) Draw a line parallel to the y-axis
Thus, the co-ordinates of another point on the circle are $(3,5)$ or $(3,-5)$
(c) Distance between the points $(3,0)$ and $(0,5)$ is given by
$d=\sqrt{{(3-0)}^2+{(0+5)}^2}=\sqrt{9+25}=\sqrt{34}>5$
Thus, the point $(0,5)$ does not lie inside the circle.
Thus, the point passes through the equation of the circle.