Question

In the figure the radius of the circle centred at C is 5. The circle passes through the point A(8, 0). If PC is perpendicular to x axis, find the coordinates of the points P, B and C.

Answer 1

Perpendicular drawn from centre of the circle on the chord of the circle bisects the cord.
CP is the perpendicular drawn from centre C on cord OX.
$OP=\frac{1}{2}OA=\frac{1}{2}\sqrt{8^2+0^2}=4$
$OP=4$
Point P lies on x axis so its y coordinate is 0.
Let $(x,0)$ be the coordinates of P.
$OP=\sqrt{(x-0{)}^2+(0-0{)}^2}=4$
$x^2=16$
$x=4$
Coordinates of point P are $(4,0)$.
In $△OPC$ using pythagoras theorem
$O{C}^2=C{P}^2+O{P}^2$
$C{P}^2=O{C}^2-O{P}^2$
$C{P}^2=5^2-4^2$
$CP=3$
Coordinates of point C are $(4,3).$
Draw CE perpendicular on chord OB.
Then y coordinate of point E and C are same.
Also x coordinate of point E is 0.
So coordinates of point E are (0,3).
Perpendicular drawn from centre of the circle on the chord of the circle bisects the chord.
$OB=2OE=\sqrt{0^2+3^2}=6$
$OB=6$
Point B lies on y axis so its x coordinate is 0.
Let $(0,y)$ be the coordinates of B.
$OB=\sqrt{(0-0{)}^2-(y-0{)}^2}=6$
$y^2=36$
$y=6$
Coordinates of point B are $(0,6).$