I need math help with tangent lines and circles!?!?

The equation of the circle which is centered at (-4,-6) and which has the line -x+3y+7=0 as a tangent is

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  • 1 decade ago
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    The equation of the circle is (x+4)^2 + (y+6)^2 = r^2 where r is to be determined.

    r is just the distance from (-4,-6) to the line x+3y+7= 0

    This distance is given by the formula:

    d = |ax1+by1+c|/sqrt(a^2+b^2), where a = 1, b= 3 and c = 7 and x1 = -4 and y1 = -6.

    So d = r = |(1)((-4) + (3)(-6) + 7|/sqrt(1^2 + 3^2)

    r = |-11|/sqrt(10)

    So r^2 = 121/10 = 12.1

    The equation is (x+4)^2 + (y+6)^2 = 12.1

  • 1 decade ago

    For this one you do not know the radius.. You have to find it.

    let r = radius -- a constant

    let (x2,y2) be the point on the line that is tangent to the circle

    Now you need to find the equation of a circle:

    (x+4)^2 + (y+6)^2 = r^2

    You are given the tangent line:

    3y = x-7

    y = x/3 - 7/3

    You know that a tangent line is always perpendicular to a line from the center of the circle to the point of tangent.

    Given the tangent line, find a line that is perpendicular to it which contains the point (-4,-6).

    Since the tangent line:

    y = x/3 - 7/3 , has slope of 1/3 a perpendicular slope is -3

    Now:

    (y-y1)= m(x-x1)

    (y+6) = (-3)(x+4)

    y = -3x -18 -- is a line that is perpendicular to the tangent and contains the mid point of the circle.

    Where these two lines intersect is the point where the tangent line has a point on the circle!

    Solving for x and y:

    x = -47/10

    y = -39/10

    All that is left to do is find the distance between the midpoint and

    the tangent point. This is your r value.

    The rest I am sure you can figure out.

    EDIT:

    I Found r^2 = 49/10 so the equation comes out to:

    (x+4)^2 + (y+6)^2 = 49/10

    r^2 = 49/10

    r = 7/sqrt(10) = 7*sqrt(10)/10 -- this is the most correct way of writing it.

    remeber for circle:

    y = +/- sqrt((49/10)-(x+4)^2) -6

    I graphed out all the lines and it made sense. You can check yourself by using this program and typing in:

    http://www.walterzorn.com/grapher/grapher_e.htm

    Type in the following (or copy paste):

    sqrt((49/10)-(x+4)^2) -6;

    -sqrt((49/10)-(x+4)^2) -6;

    x/3 -7/3;

    -3x -18;

    Those are all the graphs

  • fcas80
    Lv 7
    1 decade ago

    Distance from point (xo,yo) to line Ax + By + C = 0 is given by

    | Axo + Byo + C| / SQRT(A^2 + B^2).

    |1*-4 + 3*-6 +7| / SQRT(1^2 + 3^2) = 4.74

    So 4.74 is radius.

    Circle is (x+4)^2 + (y+6)^2 = 4.74^2

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