The equation of the circle that passes through the given points and has its center lying on this line 2x + y = 7 is equal to x² + y² - 10x + 6y - 6 = 0.
Mathematically, the standard form of the equation of a circle is given by;
(x - h)² + (y - k)² = r²
Where:
Assuming the center of this circle lies at (h, k), then we have the following expressions that models the distance from the center of the given circle:
(h - 7)² + (k - 3)² = (h - 11)² + (k - (-1))²
(h - 7)² + (k - 3)² = (h - 11)² + (k + 1)²
(h - 7)² - (h - 11)² = (k + 1)² - (k - 3)²
(h - 7 + h - 11)(h - 7 - h + 11) = (k + 1 + k - 3)(k + 1 - k + 3)
(2h - 18)(4) = (2k - 2)(4)
8h - 72 = 8k - 8
8h - 8k = 64
h - k = 8 .......equation 1.
Since (h, k) lies at 2x + y = 7, we have:
2h + k = 7 .......equation 2.
Solving eqn. 1 and eqn. 2 simultaneously, we have:
h - k = 8
2h + k = 7
3h = 15
h = 15/3
h = 5.
For the value of k, we have:
h - k = 8
5 - k = 8
k = 5 - 8
k = -3.
Next, we would determine the radius of this circle:
Radius, r² = (h - x)² + (k - y)²
Radius, r² = (5 - 7)² + (-3 - 3)²
Radius, r² = (-2)² + (-6)²
Radius, r² = 4 + 36
Radius, r = √40.
Now, we can write the equation of this circle:
(x - h)² + (y - k)² = r²
(x - 5)² + (y - (-3))² = 40
(x - 5)² + (y + 3)² = 40.
x² - 5x - 5x + 25 + y² + 3y + 3y + 9 - 40 = 0
x² - 10x + 25 + y² + 6y - 31 = 0
x² - 10x + y² + 6y - 6 = 0
x² + y² - 10x + 6y - 6 = 0.
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