a) [tex]m=\frac{-12-4}{5-(-7)}=\frac{-16}{12}=\boxed{-\frac{4}{3}}[/tex]
b) We know the equation is [tex]y=-\frac{4}{3}x+b[/tex], where b is a constant. If we substitute in the coordinates of point a, we get that:
[tex]4=-\frac{4}{3}(-7)+b\\4=\frac{28}{3}+b\\b=-\frac{16}{3}[/tex]
This means the y-intercept is [tex]\boxed{\left(0, -\frac{16}{3} \right)}[/tex]
To find the x-intercept, we can set the equation equal to 0 and solve for x.
[tex]0=-\frac{4}{3}x-\frac{16}{3}\\\frac{16}{3}=-\frac{4}{3}x\\x=-4[/tex]
So, the x-intercept is [tex]\boxed{(-4, 0)}[/tex]
c) [tex]\left(\frac{-7+5}{2}, \frac{4-12}{2} \right)=\boxed{(-1, -4)}[/tex]
d) [tex]ab=\sqrt{(-7-5)^{2}+(-12-4)^{2}}=\boxed{20}[/tex]
e) The perpendicular bisector must pass through the midpoint, (-1, 4), and have a slope that is the negative reciprocal (so in this case, 3/4). Substituting into point-slope form,
[tex]y-4=\frac{3}{4}(x+1)\\y-4=\frac{3}{4}x+\frac{3}{4}\\\boxed{y=\frac{3}{4}x+\frac{19}{4}}[/tex]
f) If ab is a diameter of said circle, this means it is centered at the midpoint, (-1, 4), and has a radius of 20/2=10. This means the equation is
[tex]\boxed{(x+1)^{2}+(y-4)^{2}=100}[/tex]
