A circle has a center that falls on the line $y = \frac{2}{9} x + 8$ $y = 2/9x +8$ and passes through $(3, 1)$ $( 3 ,1 )$ and $(5, 7)$ $(5 ,7 )$ . What is the equation of the circle?

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Answer 1 · 2018-01-04T12:00:43

The equation of circle is $(x + 24/5)^2 + (y – 104/15)^2 = 9.8^2$

The center-radius form of the circle equation is

$(x – h)^2 + (y – k)^2 = r^2$ , with the center being at the point

$(h, k)$ and the radius being $r$ . The points $(3,1) and (5,7)$

are on the circle. $:. (3 – h)^2 + (1 – k)^2 =(5 – h)^2 + (7 – k)^2$

$:. 9 – 6h+cancelh^2 + 1 – 2k+cancelk^2 =25 – 10h+cancelh^2 +49 – 14k+cancelk^2$

$:. 9 – 6h + 1 – 2k =25 – 10h +49 – 14k$ or

$10h – 6h + 14k – 2k+ =25+49-9 – 1$ or

$4h+12k=64 or h+3k=16 (1); (h,k)$ lies on the straight line

$y=2/9x+8 :. k = 2/9h+8 or 9k-2h=72 (2)$ . Multiplying

the equation (1) by $2$ we get $2h+6k=32 (3)$ . Adding

equation (2) and equation (3) we get $15k=104 or k=104/15$

Putting $k=104/15$ in equation (1) we get $h=16-3*104/15$ or

$h=16-104/5 or h = (80-104)/5=-24/5 :.$ Centre is at

$(h,k) or (-24/5 , 104/15) :. r^2= (3 + 24/5)^2 + (1 – 104/15)^2$

or $r^2~~96.04 or r~~ 9.8$ . Therefore the equation of circle is

$(x + 24/5)^2 + (y – 104/15)^2 = 9.8^2$ [Ans]

A circle has a center that falls on the line y = 2/9x +8 y=29x+8y = 2/9x +8 and passes through ( 3 ,1 )(3,1) ( 3 ,1 ) and (5 ,7 )(5,7)(5 ,7 ). What is the equation of the circle?