The coordinates of the centre of a circle are (x − 7, 2x). Find the value(s) of ‘x’, if the circle passes through the point (−9, 11) and has radius 5√2 units.
For a circle,
distance between centre and any point on circle = radius
using square of distance formula:
(x₂ − x₁)² + (y₂ − y₁)² = r²
Given
Centre = (x − 7, 2x)
Point on circle = (−9, 11)
Radius r = 5√2
r² = (5√2)² = 25 × 2 = 50
(-9 - (x - 7))² + (11 - 2x)² = 50
(-9 - x + 7)² + (11 - 2x)² = 50
(-x - 2)² + (11 - 2x)² = 50
(x + 2)² + (11 - 2x)² = 50
Expand using identities
(x² + 4x + 4) + (121 - 44x + 4x²) = 50
5x² - 40x + 125 = 50
5x² - 40x + 125 - 50 =0
5x² - 40x + 75 = 0
Divide by 5
x² - 8x + 15 = 0
Factorise:
x² - 5x - 3x + 15 = 0
x(x - 5) - 3(x - 5) = 0
(x - 5)(x - 3) = 0
x = 5 or x = 3
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