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Math Help - circle, tangent, point problems?????

  1. #1
    Member sluggerbroth's Avatar
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    circle, tangent, point problems?????

    How do you solve these problems?
    1) find standard equation of circles that pass thru (2,3) and are tangent to both lines 3x-4y=-1 and 4x+3y=7
    2 find standard eqations of circles that have centers on 4x+3y=8 and are tangent to both the line x+y=-2 and 7x-y=-6
    3) find eqations of lines thru (4,10) and tangent to circle x^2+y^2-4y-36=0

    these problems come at the beginning of calculus 1???? help!!!!!!!!
    Last edited by sluggerbroth; April 21st 2012 at 06:13 PM. Reason: calc1
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    Re: circle, tangent, point problems?????

    Quote Originally Posted by sluggerbroth View Post
    How do you solve these problems?
    1) find standard equation of circles that pass thru (2,3) and are tangent to both lines 3x-4y=-1 and 4x+3y=7
    2 find standard eqations of circles that have centers on 4x+3y=8 and are tangent to both the line x+y=-2 and 7x-y=-6
    3) find eqations of lines thru (4,10) and tangent to circle x^2+y^2-4y-36=0
    Hint: Tangents have the same gradient as curves at the point at which they are tangent.
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    Member sluggerbroth's Avatar
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    Re: circle, tangent, point problems?????

    these problems come at the beginning of calculus 1???? help!!!!!!!!
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    Re: circle, tangent, point problems?????

    Quote Originally Posted by sluggerbroth View Post
    these problems come at the beginning of calculus 1???? help!!!!!!!!
    For the first question, what is the gradient of each of your tangents?
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    Re: circle, tangent, point problems?????

    One method of solving problem 1 comes from analytic geometry. As you can check, the lines 3x - 4y = -1 and 4x + 3y = 7 are perpendicular and intersect at (1, 1). Let us consider two coordinate systems with the origin (1, 1): one has axes parallel to the original axes (the latter are drawn dashed in the figure below) and the other has the lines 3x - 4y = -1 and 4x + 3y = 7 as axes. The point A(2, 3) in the original (dashed) system has coordinates x_0 = 1 and y_0 = 2 in the new (x, y)-system.



    The circle that touches x'- and y'-axes must have its center O on the angle bisector that has the equation y' = x'. In addition, the distance OC between B and the x'-axis must equal AO. (In this problem, AO happens to be parallel to the x'-axis, but this is a coincidence.) How do we find the coordinates of O? Recall that the locus of points equidistant from the point A and the x'-axis is a parabola. It is easier to find its equation in the system (x'', y') where the y'-axis is shifted so that it goes through A (dashed line). Then x' = x'' + x_0'. If O has coordinates (x'', y'), then AO^2 = x''^2+(y'-y_0')^2 and OC=y'^2. So, x''^2+(y'-y_0')^2=y'^2 is the equation of the parabola. The equation of the bisector in the (x'', y')-system is y' = x'' + 2. Solving these two equations allows finding x'' (and hence x') and y' knowing x_0' and y_0'.

    We know the coordinates (x_0=1, y_0=2) of A in the (x, y)-system. How do we find the coordinates (x_0',y_0') of the same point in the (x', y')-system? Suppose that (x_1,y_1) and (x_2,y_2) are unit vectors along the x'-, y'-axes expressed in terms of x and y. Then

    \binom{x_0}{y_0}=x_0'\binom{x_1}{y_1}+y_0'\binom{x  _2}{y_2}

    i.e.,

    x_0=x_1x_0'+x_2y_0'
    y_0=y_1x_0'+y_2y_0' (*)

    This allows finding (x_0,y_0) from (x_0',y_0'), which is also needed below. To find (x_0',y_0') from (x_0,y_0) there are several ways. First, one can solve the system (*) for x_0',y_0'. Second, one can use the formula for the distance of a point (x_0,y_0) from a line ax+by+c. This distance is \frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}.

    So, here are the instructions.

    Step 1. Find the unit vectors (x_1,y_1) and (x_2,y_2) along the x'- and y'-axes.

    Step 2. Find the coordinates (x_0',y_0') of A in the (x', y')-system.

    Step 3. Find the equation of the parabola in the (x'', y')-system.

    Step 4. Find the coordinates of the circle center O in the (x'', y')-system.

    Step 5. Convert the coordinates of O into the (x', y')-system and then (x, y)-system using (*).

    Step 6. Find the radius OA = OC of the circle and write the circle equation in the (x, y)-system.
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