OP : PQ : OQ = 4 : 3 : 5

OP = 8,
so PQ = 6
coordinates of Q is (8, 6)

(ii)
OQ is diameter,
radius of circle is 2.5 units.

midpoint M of OQ is (4, 3)
which is centre of circle.

equation of circle:
(x - 4)² + (y - 3)² = (2.5)²
4(x - 4)² + 4(y - 3)² = 25
(2x - 8)² + (2y - 6)² = 25

OR
x² - 8x + 16 + y² - 6y + 9 - 6.25 = 0
x² - 8x + y² - 6y + 18.75 = 0
4x² - 32x + 4y² - 24y + 75 = 0


(iii)

if x-coordinate of R is 8,
which is directly above (or below) P,
(y-coordinate of R is unknown);

tan ∡ORP = tan 38°
RP = 10.24 units (n.a.)
as 10.24 units is more than 6 units,

which means R is above pt Q
OR
R is below pt P, which is outside the circle;
so x-coordinate of R is not equal to 8.

since R lies inside the circle (given),
then x-coordinate of R has to be less than 8.

' tan POQ = 0.75' is just another way of saying 'the gradient of OQ is 0.75' (since OP is horizontal and Q is directly above P'.

'lies on the circle' probably means somewhere on the whole circle (inside included) instead of just some point on its circumference

Usually in the exams, as far as I can remember, ‘on the circle’ strictly means ‘on the exact outline/circus of the circle’.

likely a phrasing issue then. The setter of the question has to rectify

But even then R must be outside the circle. Draw R inside the triangle and for all combis, angle ORP cannot be 38.

If R was outside the circle, it defeats the purpose of asking whether the x-coordinate is more or less than 8 , since both are possible.

From the way this part of the question is set,The setter probably intended for R to be inside the circle. it is likely that the value of 38° is more wrong than the phrasing.

Angle OQP is 53 degrees though upon calculation.

For angle ORP, if R is ON the circle, it will be 53 degrees also. The closer R is to the centre, the greater the angle would be (by the properties of the circle); if R is the centre, then angle ORP would be 106 degrees.

It’s not possible for R to be inside the circle as a result.

Possibly the question should not exist. The 38 degrees was likely an error.

The only way the question would make sense is to change ORP to a value higher than 53.1°, and change the phrasing from ' lies on' to 'lies within'