Explanation:
the equation of a circle is
(x – h)² + (y – k)² = r²
where (h, k) are the coordinates of the center of the circle, and r is the radius of the circle.
now, the center of the circle is the point (h, k) on the given line that has the same distance (which is then r) to the given 2 points.
the line is
x - 10y = 14
let's convert this to the slope-intercept form
y = ax + b
"a" is the slope, "b" is the y- intercept of the line.
x = 10y + 14
10y = x - 14
y = x/10 - 14/10 = x/10 - 7/5
k = h/10 - 7/5
so the center of the circle is
(h, h/10 - 7/5)
the distance² (r²) of this point to the 2 given points is then for (10, 2) :
(10 - h)² + (2 - (h/10 - 7/5))² = (10 - h)² + (17/5 - h/10)²
for (9, -3) :
(9 - h)² + (-3 - (h/10 - 7/5))² = (9 - h)² + (-8/5 - h/10)²
remember, both must be equal (have the same result) : r².
the first is
100 - 20h + h² + 289/25 - 34h/50 + h²/100 =
= 10000/100 - 2000h/100 + 100h²/100 + 1156/100 - 68h/100 + h²/100 =
= 11156/100 - 2068h/100 + 101h²/100
the second is
81 - 18h + h² + 64/25 + 16h/50 + h²/100 =
= 8100/100 - 1800h/100 + 100h²/100 + 256/100 + 32h/100 + h²/100 =
= 8356/100 - 1768h/100 + 101h²/100
since both must be equal :
11156/100 - 2068h/100 + 101h²/100 =
= 8356/100 - 1768h/100 + 101h²/100
11156/100 - 2068h/100 = 8356/100 - 1768h/100
11156 - 2068h = 8356 - 1768h
2800 = 300h
28 = 3h
h = 28/3 = 9.333333...
k = h/10 - 7/5 = 28/3/10 - 7/5 = 28/30 - 7/5 =
= 14/15 - 7/5 = 14/15 - 21/15 = -7/15 = -0.4666666...
for r² let's use e.g. the first given point :
(10 - h)² + (2 - k)² = r²
(10 - 28/3)² + (2 - -7/15)² = r²
(30/3 - 28/3)² + (30/15 + 7/15)² = r²
(2/3)² + (37/15)² = r²
(10/15)² + (37/15)² = r²
100/225 + 1369/225 = r²
1469/225 = r²
r = sqrt(1469)/15
so, the circle equation is
(x - 28/3)² + (y + 7/15)² = 1469/225