First calculate the slope:

\(\displaystyle (\text{change in Y})/(\text{change in x}) = (1-2)/(3-2) = -1\)

The standard equation for a line is  \(\displaystyle Y=a*X + b\).

In this equation, \(\displaystyle a\) is the slope of the line, and \(\displaystyle b\) is the \(\displaystyle y\)-intercept. All points on the line must fit this equation. Plug in either point (1,3) or (2,2).

Plugging in (1,3) we get \(\displaystyle 3=-1*1 + b\).

Therefore, \(\displaystyle b = 3 - (-1) =4\).

Our equation for the line is now:

\(\displaystyle Y= (-1)X + 4\)

To find the \(\displaystyle x\)-intercept, we plug in \(\displaystyle y=0\):

\(\displaystyle 0=(-1)x+4\) 

 \(\displaystyle x=4\)

Thus, the \(\displaystyle x\)-intercept the point (4,0).

To find the y-intercept, let \(\displaystyle x\) equal 0.

\(\displaystyle 3(0)+6y=15\)

We can then solve for the value of \(\displaystyle y\).

\(\displaystyle 6y=15\)

\(\displaystyle y=\frac{15}{6}=2.5\)

The y-intercept will be \(\displaystyle (0,2.5)\).

To find the x-intercept, set y equal to zero and solve:

\(\displaystyle 2y=3x+5\)

\(\displaystyle 2(0)=3x+5\)

\(\displaystyle 0=3x+5\)

Subtract \(\displaystyle 5\) from both sides:

\(\displaystyle -5=3x\)

Divide both sides by \(\displaystyle 3\) to isolate x:

\(\displaystyle \frac{-5}{3}=x\)

\(\displaystyle y=\frac{2}{3}x+ 12\)

To find the y-intercept, we set the \(\displaystyle x\) value equal to zero and solve for the \(\displaystyle \small y\) value.

\(\displaystyle y= \frac{2}{3} (0) +12\)

\(\displaystyle y=12\)

Since the y-intercept is a point, we will need to convert our answer to point notation.

\(\displaystyle (0,12)\)

First, we find the slope between the two points:

\(\displaystyle P_{1}=(-3,-7)\) and \(\displaystyle P_{2}=(2,3)\)

\(\displaystyle m= \frac{y_{2} - y_{1}}{x_{2}-x_{1}}=\frac{3-(-7)}{2-(-3)}=\frac{10}{5}=2\)

Plug the slope and one point into the slope-intercept form to calculate the intercept:

\(\displaystyle y=mx+b\) 

\(\displaystyle 3=2(2)+b\) 

\(\displaystyle b=-1\)

Thus the equation between the points becomes \(\displaystyle y=2x-1\).

The general equation of a circle is \(\displaystyle (x - h)^{2} + (y - k)^{2} = r^{2}\), where the center of the circle is \(\displaystyle (h, k)\) and the radius is \(\displaystyle r\).

Thus, we plug the values given into the above equation to get \(\displaystyle (x - 2)^{2} + (y - 4)^{2} = 100\). 

The standard formula for a circle is \(\displaystyle (x-a)^2+(y-b)^2=r^2\), with \(\displaystyle (a,b)\) the center of the circle and \(\displaystyle r\) the radius.

Plug in our given information.

\(\displaystyle (x-a)^2+(y-b^2)=r^2\)

\(\displaystyle (x-2)^2+(y-3)^2=25\)

This describes what we are looking for.  This equation is not one of the answer choices, however, so subtract \(\displaystyle 25\) from both sides.

\(\displaystyle (x-2)^2+(y-3)^2-25=0\)