Points $A(-1, y)$ and $B(5, 7)$ lie on a circle with centre $O(2, -3y)$ such that $AB$ is a diameter of the circle. Find the value of $y$. Also, find the radius of the circle.
Generated by claude-sonnet-4-6 · 2026-06-15 10:29 · grounding rag
Model Answer
Since AB is a diameter and O is the centre, O is the midpoint of AB.
Using the midpoint formula:
$$\frac{-1+5}{2} = 2 \quad \checkmark$$
$$\frac{y+7}{2} = -3y$$
$$y + 7 = -6y$$
$$7y = -7 \implies y = -1$$
So A = (−1, −1) and B = (5, 7).
Radius = $\frac{1}{2}AB = \frac{1}{2}\sqrt{(5-(-1))^2+(7-(-1))^2} = \frac{1}{2}\sqrt{36+64} = \frac{1}{2}\times 10 = 5$ units.
y = −1; Radius = 5 units.
Source: Chapter 7, Section 7.3 (Midpoint Formula) and Section 7.2 (Distance Formula)
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Explanation
- Since O is the centre of a circle with diameter AB, O must be the midpoint of AB — this is the key property to use.
- Apply the midpoint formula to the y-coordinates only (x-coordinate is automatically satisfied) to get a linear equation in y.
- Then substitute y = −1 to find coordinates of A and B, and use the distance formula for AB; radius = AB/2.
- Examiners award 1 mark for finding y and 1 mark for the radius.