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?

Answer 1

#(x-147/38)^2+(y -101/38)^2=(sqrt[4505/2]/19)^2#

The circle equation is #C->(x-x_c)^2+(y-y_c)^2=r^2#. the straight #y = 3/7x+1# can be written as #-3x+7(y-1)=0# or #(p-p_0).vec v=0# where #p = (x,y), p_0=(0,1)# and #vec v = (-3,7)#
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 #(7,3)# orthogonal to #vec v#. The circle center given by #p_c=(x_c,y_c)# is equidistant from #p_1=(2,1)# and #p_2 = (3,5)#
So we can equate #norm(p_c-p_1) = norm (p_c-p_2)# but #p_c = p_0+lambda_c vec v^T# so we can state: #(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_1-2p_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_1-2p_0.(p_2-p_1)-2lambda_c vec v^T.(p_2-p_1)=0# and finally #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 #lambda_c = 21/38 # then #p_c = (147/38,101/38)#
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Answer 2

The equation of the circle is ( (x - 17/13)^2 + (y - 37/13)^2 = 2/13 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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