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Find equation of circle passes through points (10,2) , (9,-3) and center on the line x-10y=14​

User Chlunde
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1 Answer

7 votes

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

User Ante
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