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

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

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Answer 1 · 2016-05-17T15:51:02

${(x - \frac{147}{38})}^{2} + {(y - \frac{101}{38})}^{2} = {⎛ ⎜ ⎜ ⎝ \frac{\sqrt{\frac{4505}{2}}}{19} ⎞ ⎟ ⎟ ⎠}^{2}$ $(x-147/38)^2+(y -101/38)^2=(sqrt[4505/2]/19)^2$

Explanation:

The circle equation is
$C \to {(x - x_{c})}_{c}^{2} + {(y - y_{c})}_{c}^{2} = r^{2}$ $C->(x-x_c)^2+(y-y_c)^2=r^2$ .
the straight $y = \frac{3}{7} x + 1$ $y = 3/7x+1$ can be written as
$- 3 x + 7 (y - 1) = 0$ $-3x+7(y-1)=0$ or $(p - p_{0}) . \to v = 0$ $(p-p_0).vec v=0$ where
$p = (x, y), p_{0} = (0, 1)$ $p = (x,y), p_0=(0,1)$ and $\to v = (- 3, 7)$ $vec v = (-3,7)$

The parametric representation for the straight line
is given by $p = p_{0} + λ {\to v}^{T}$ $p = p_0 + lambda vec v^T$ where ${\to v}^{T}$ $vec v^T$ is the vector with components $(7, 3)$ $(7,3)$ orthogonal to $\to v$ $vec v$ . The circle center given by $p_{c} = (x_{c}, y_{c})$ $p_c=(x_c,y_c)$ is equidistant from $p_{1} = (2, 1)$ $p_1=(2,1)$ and $p_{2} = (3, 5)$ $p_2 = (3,5)$

So we can equate
$∥ p_{c} - p_{1} ∥ = ∥ p_{c} - p_{2} ∥$ $norm(p_c-p_1) = norm (p_c-p_2)$ but $p_{c} = p_{0} + λ_{c} {\to v}^{T}$ $p_c = p_0+lambda_c vec v^T$ so we can state:
$(p_{0} + λ_{c} {\to v}^{T} - p_{1}) . (p_{0} + λ_{c} {\to v}^{T} - p_{1}) = (p_{0} + λ_{c} {\to v}^{T} - p_{1}) . (p_{0} + λ_{c} {\to v}^{T} - p_{1})$ $(p_0+lambda_c vec v^T-p_1).(p_0+lambda_c vec v^T-p_1)=(p_0+lambda_c vec v^T-p_1).(p_0+lambda_c vec v^T-p_1)$ .
Developing and grouping
$p_{1} . p_{1} - 2 p_{0} . p_{1} - 2 λ_{c} {\to v}^{T} . p_{1} = p_{2} . p_{2} - 2 p_{0} . p_{2} - 2 λ_{c} {\to v}^{T} . p_{2}$ $p_1.p_1-2p_0.p_1-2lambda_c vec v^T.p_1 = p_2.p_2-2p_0.p_2-2lambda_c vec v^T.p_2$
or
$p_{2} . p_{2} - p_{1} . p_{1} - 2 p_{0} . (p_{2} - p_{1}) - 2 λ_{c} {\to v}^{T} . (p_{2} - p_{1}) = 0$ $p_2.p_2-p_1.p_1-2p_0.(p_2-p_1)-2lambda_c vec v^T.(p_2-p_1)=0$
and finally
$λ_{c} = \frac{p_{2} . p_{2} - p_{1} . p_{1} - 2 p_{0} . (p_{2} - p_{1})}{2 {\to v}^{T} . (p_{2} - p_{1})}$ $lambda_c = (p_2.p_2-p_1.p_1-2p_0.(p_2-p_1))/(2 vec v^T.(p_2-p_1))$
Substituting values we obtain $λ_{c} = \frac{21}{38}$ $lambda_c = 21/38$ then $p_{c} = (\frac{147}{38}, \frac{101}{38})$ $p_c = (147/38,101/38)$

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

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