1. Find the equation of the diameter of the circle x^2 + y^2 - 4x + 6y = 14 which passes through the origin.
2. Find the equation of the diameter of the circle x^2 + y^2 -3x + 2y = 26 which cuts the x-axis at an angle of 45 degrees.
to #1.:
Calculate the coordinates of the centre of the circle by completing the squares:
$\displaystyle x^2-4x {\color{red}+4} +y^2 + 6y {\color{red} + 9} = 14 {\color{red} +4 +9}~\implies~ (x-2)^2+ (y+3)^2=27$
The line in question has to pass through (2, -3) and the origin.
You should get: $\displaystyle y = -\frac32 x$
to #2:
Calculate the coordinates of the centre of the circle by completing the squares: $\displaystyle \left(x-\frac32\right)^2+(y+1)^2=\frac{117}4$
There are two cases possible if the x-axis and a straight line include an angle of 45°: The slope of the line can be m = 1 or m = -1
Use the point-slope-formula to calculate the equation of the line:
a) m = 1:
$\displaystyle y-1=\left(x-\frac32\right)~\implies~ \boxed{y=x-\frac12}$
b) m = -1
$\displaystyle y-1=-\left(x-\frac32\right)~\implies~ \boxed{y=-x+\frac52}$