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Math Help - circle geomtery help

  1. #1
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    circle geomtery help

    The circle C has centre (5, 2) and passes through the point (7, 3).

    (a) Find the length of the diameter of C.
     2x \sqrt{4+1} = 2\sqrt{5}



    (b) Find an equation for C.
     (x-5)^2 + (y-2)^2 = 5

    (c) Show that the line y = 2x − 3 is a tangent to C and find the coordinates
    of the point of contact.

    Need help with question 'c'.

    I know that the angle between a tangent and radius is 90 , but how would I show that the product of their gradients is -1, if I dont know the point of contact?

    Also how do I find the point were it touches the circle?
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  2. #2
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    Quote Originally Posted by Tweety View Post
    The circle C has centre (5, 2) and passes through the point (7, 3).

    (a) Find the length of the diameter of C.
     2x \sqrt{4+1} = 2\sqrt{5}



    (b) Find an equation for C.
     (x-5)^2 + (y-2)^2 = 5

    (c) Show that the line y = 2x − 3 is a tangent to C and find the coordinates
    of the point of contact.

    Need help with question 'c'.

    I know that the angle between a tangent and radius is 90 , but how would I show that the product of their gradients is -1, if I dont know the point of contact?

    Also how do I find the point were it touches the circle?
    Here's one approach out of many possible approaches: Show that the two equations

    y = 2x - 3

     (x-5)^2 + (y-2)^2 = 5

    have only one solution.
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  3. #3
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    Quote Originally Posted by mr fantastic View Post
    Here's one approach out of many possible approaches: Show that the two equations

    y = 2x - 3

     (x-5)^2 + (y-2)^2 = 5

    have only one solution.
    So if I solved this simutaneously, wouldn't that mean that these to equations intersect at that one point, instead of touching?
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  4. #4
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    Quote Originally Posted by Tweety View Post
    So if I solved this simutaneously, wouldn't that mean that these to equations intersect at that one point, instead of touching?
    If a line intersects a circle at one point then it is tangent to the circle at that point.
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