(i) Given the points A (1, −8), B (−4, 7) and C (−7, −4), show that they lie on a circle K whose center is the origin O (0, 0). What is the radius of circle K?
(ii) Given the points D (−5, 6) and E (0, 9), check whether D and E lie within the circle, on the circle, or outside the circle K.
- (i) Given the points A (1, −8), B (−4, 7) and C (−7, −4), show that they lie on a circle K whose center is the origin O (0, 0). What is the radius of circle K?
- (ii) Given the points D (−5, 6) and E (0, 9), check whether D and E lie within the circle, on the circle, or outside the circle K.
Step-by-step solution
Idea: Every point of a circle is the same distance (the radius) from the centre. So compare each point’s distance from O with the radius: equal means on the circle, less means inside, more means outside.
(i) Given the points A (1, −8), B (−4, 7) and C (−7, −4), show that they lie on a circle K whose center is the origin O (0, 0). What is the radius of circle K?
- OA = √(12 + (−8)2) = √(1 + 64) = √65.½ mark
- OB = √((−4)2 + 72) = √(16 + 49) = √65.½ mark
- OC = √((−7)2 + (−4)2) = √(49 + 16) = √65.½ mark
- All three are √65 from O, so they lie on the circle K with centre O and radius √65 units (about 8.06 units).½ mark
(ii) Given the points D (−5, 6) and E (0, 9), check whether D and E lie within the circle, on the circle, or outside the circle K.
- OD = √((−5)2 + 62) = √(25 + 36) = √61. Since 61 < 65, OD < radius, so D lies within (inside) the circle.1 mark
- OE = √(02 + 92) = 9 = √81. Since 81 > 65, OE > radius, so E lies outside the circle.1 mark
Check: √65 ≈ 8.06. E is on the y-axis at height 9, beyond the circle’s top point (0, 8.06) ✓. √61 ≈ 7.81 < 8.06 ✓.
Answer to write in the exam
(i)
OA = √(12 + 82) = √65
OB = √(42 + 72) = √65
OC = √(72 + 42) = √65
OA = OB = OC ⇒ A, B, C lie on a circle with centre O
∴ Radius of K = √65 units
(ii)
OD = √(52 + 62) = √61 < √65 ∴ D lies inside circle K
OE = √(0 + 92) = 9 = √81 > √65 ∴ E lies outside circle K
Common mistakes that cost marks
- Adding the coordinates instead of squaring: OA = 1 + 8 = 9. Use √(x2 + y2).
- Giving the radius as 65. The radius is √65; 65 is the radius squared.
- Taking (−8)2 = −64, which gives OA2 = −63. Squares are never negative.
How this can come in the exam
The point (6, −7) lies ______ the circle with centre O (0, 0) and radius √85.
- inside
- on
- outside
- at the centre of
Show answer
(B) on
Distance from O = √(36 + 49) = √85, which equals the radius.
The point (k, 3) lies on the circle with centre O (0, 0) and radius 5. Find the possible values of k.
Show answer
k2 + 32 = 52 (1 mark), so k2 = 16 and k = 4 or −4 (1 mark).Try one yourself
Does P (−6, 6) lie inside, on or outside circle K (centre O, radius √65)? What about Q (8, −1)?
Show answer
OP2 = 36 + 36 = 72 > 65, so P is outside. OQ2 = 64 + 1 = 65, so Q is on the circle.
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