A circle has a center that falls on the line $y = \frac{7}{6} x + 1$ $y = 7/6x +1$ and passes through $(9, 4)$ $(9 ,4 )$ and $(8, 5)$ $(8 ,5 )$ . What is the equation of the circle?

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A circle has a center that falls on the line $y = \frac{7}{6} x + 1$ $y = 7/6x +1$ and passes through $(9, 4)$ $(9 ,4 )$ and $(8, 5)$ $(8 ,5 )$ . What is the equation of the circle?

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Answer 1 · 2016-05-18T14:41:24

${(x + 31)}^{2} + {(y + 35)}^{2} = 3121$ $(x+31)^2+(y+35)^2=3121$

Explanation:

The straight $y = \frac{7}{6} x + 1$ $y = 7/6x+1$ can be represented as $(p - p_{0}) . \to v = 0$ $(p-p_0).vec v =0$
where $p = (x, y), p_{0} = (- 1, 0)$ $p = (x,y), p_0 = (-1,0)$ and $\to v = (7, - 6)$ $vec v = (7,-6)$ and also can be represented in the parametric form as:

$S_{1} \to p = p_{0} + λ \cdot {\to v}^{T}$ $S_1-> p = p_0+lambda* vec v^T$ where ${\to v}^{T}$ $vec v^T$ is the vector with components $(6, 7)$ $(6,7)$ which is orthogonal to $\to v$ $vec v$ .

The center point to the circle is at the intersection of $S_{1}$ $S_1$ and $S_{2}$ $S_2$ where $S_{2}$ $S_2$ is the mediatrix to the segment $¯ ¯¯¯¯¯ ¯ p_{1} p_{2}$ $bar(p_1p_2)$ . $S_{2}$ $S_2$ is written as:
$S_{2} \to p = p_{12} + μ \cdot {\to v}_{12}^{T}$ $S_2->p = p_{12}+mu*vec v_{12}^T$ where $p_{12} = \frac{p_{1} + p_{2}}{2}, {\to v}_{12} = p_{1} - p_{2} = (1, - 1)$ $p_{12} = (p_1+p_2)/2,vec v_{12} = p_1-p_2 = (1,-1)$ and ${\to v}_{12}^{T} = {- 1, - 1}$ $vec v_{12}^T = {-1,-1}$ .

Solving $p_{0} + λ_{c} \cdot {\to v}^{T} = p_{12} + μ_{c} \cdot {\to v}_{12}^{T}$ $p_0+lambda_c* vec v^T = p_{12}+mu_c*vec v_{12}^T$ for $μ_{c}, λ_{c}$ $mu_c, lambda_c$ we obtain $μ_{c} = - 5, λ_{c} = \frac{79}{2}$ $mu_c= -5, \lambda_c = 79/2$
Now, putting all together,
$p_{c} = p_{0} + λ_{c} \cdot {\to v}^{T} = (- 31, - 35)$ $p_c = p_0+lambda_c* vec v^T =(-31, -35)$
$r = ∥ p_{c} - p_{1} ∥ = \sqrt{3121}$ $r = norm(p_c-p_1)=sqrt[3121]$

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A circle has a center that falls on the line $y = \frac{7}{6} x + 1$ $y = 7/6x +1$ and passes through $(9, 4)$ $(9 ,4 )$ and $(8, 5)$ $(8 ,5 )$ . What is the equation of the circle?

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A circle has a center that falls on the line y=76x+1y = 7/6x +1 and passes through (9,4)(9 ,4 ) and (8,5)(8 ,5 ). What is the equation of the circle?

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A circle has a center that falls on the line $y = \frac{7}{6} x + 1$ $y = 7/6x +1$ and passes through $(9, 4)$ $(9 ,4 )$ and $(8, 5)$ $(8 ,5 )$ . What is the equation of the circle?