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

May 17, 2016

${\left(x - \frac{147}{38}\right)}^{2} + {\left(y - \frac{101}{38}\right)}^{2} = {\left(\frac{\sqrt{\frac{4505}{2}}}{19}\right)}^{2}$

#### Explanation:

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

The parametric representation for the straight line
is given by $p = {p}_{0} + \lambda {\vec{v}}^{T}$ where ${\vec{v}}^{T}$ is the vector with components $\left(7 , 3\right)$ orthogonal to $\vec{v}$. The circle center given by ${p}_{c} = \left({x}_{c} , {y}_{c}\right)$ is equidistant from ${p}_{1} = \left(2 , 1\right)$ and ${p}_{2} = \left(3 , 5\right)$

So we can equate
$\left\lVert {p}_{c} - {p}_{1} \right\rVert = \left\lVert {p}_{c} - {p}_{2} \right\rVert$ but ${p}_{c} = {p}_{0} + {\lambda}_{c} {\vec{v}}^{T}$ so we can state:
$\left({p}_{0} + {\lambda}_{c} {\vec{v}}^{T} - {p}_{1}\right) . \left({p}_{0} + {\lambda}_{c} {\vec{v}}^{T} - {p}_{1}\right) = \left({p}_{0} + {\lambda}_{c} {\vec{v}}^{T} - {p}_{1}\right) . \left({p}_{0} + {\lambda}_{c} {\vec{v}}^{T} - {p}_{1}\right)$.
Developing and grouping
${p}_{1.} {p}_{1} - 2 {p}_{0.} {p}_{1} - 2 {\lambda}_{c} {\vec{v}}^{T} . {p}_{1} = {p}_{2.} {p}_{2} - 2 {p}_{0.} {p}_{2} - 2 {\lambda}_{c} {\vec{v}}^{T} . {p}_{2}$
or
${p}_{2.} {p}_{2} - {p}_{1.} {p}_{1} - 2 {p}_{0.} \left({p}_{2} - {p}_{1}\right) - 2 {\lambda}_{c} {\vec{v}}^{T} . \left({p}_{2} - {p}_{1}\right) = 0$
and finally
${\lambda}_{c} = \frac{{p}_{2.} {p}_{2} - {p}_{1.} {p}_{1} - 2 {p}_{0.} \left({p}_{2} - {p}_{1}\right)}{2 {\vec{v}}^{T} . \left({p}_{2} - {p}_{1}\right)}$
Substituting values we obtain ${\lambda}_{c} = \frac{21}{38}$ then ${p}_{c} = \left(\frac{147}{38} , \frac{101}{38}\right)$