The Coordinates of the Centre of a Circle Are (x − 7, 2x): Find the Value of x if the Circle Passes Through (−9, 11) and Has Radius 5√2 | Step-by-Step Solution

 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

for  more explanation watch the video  

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