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Math Help - Circle equations

  1. #1
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    Circle equations

    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.
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  2. #2
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    Quote Originally Posted by delicate_tears View Post
    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:

    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: y = -\frac32 x

    to #2:

    Calculate the coordinates of the centre of the circle by completing the squares: \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:

    y-1=\left(x-\frac32\right)~\implies~ \boxed{y=x-\frac12}

    b) m = -1

    y-1=-\left(x-\frac32\right)~\implies~ \boxed{y=-x+\frac52}
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