Question

Determine the centre of the circle on which the points (1, 7), (7 – 1), and (8, 6) lie. What is the radius of the circle?

Answer 1

Let P(x, y) be the centre of the circle and A(1, 7), B(7, -1) and C(8, 6) be the given points.

Then PA = PB = PC = radius
⇒ PA² = PB = PC = r²
⇒ (x - 1)² + y( - 7)² = r²        ...(i)
(x - 7)² + (y + 1)² = r²           ...(ii)
Also (x - 8)² + (y - 6)² = r²   ...(iii)
Subtracting (ii) from (i)
(x² + 1 - 2x + y² + 49 - 14y)
-(x² + 49 - 14x + y² + 1 + 2y) = 0
⇒ 12x - 16y = 0
`y = (3)/(4)x`                               ...(iv)
Subtracting (iii) from (ii)
⇒ (x² + 49 - 14x + y² + 1 + 2y)
-(x² + 64 - 16x + y² + 36 - 12y) = 0
⇒ 2x + 14y - 50 = 0
⇒ `x + 7(3/4x)-25` = 0
⇒ `(25x)/(4)` = 25
⇒ x = 4
From (iv) y = `(3)/(4)x = (3)/(4) xx 4`
y = 3.
The centre is P(4, 3).
Also radius, r = PA = `sqrt((4 - 1)^2 + (3 - 7)^2)`
= `sqrt(9 + 16)`
= 5 units.