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Math Help - How to solve this?

  1. #1
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    How to solve this?

    hello im bit confused how to solve this

    1.Show that the line y = mx + c touches the circle x^2 + y^2 = a^2 if c^2 = a^2(1+m^2)


    2.The point (p,q) lies on the circle x^2 + y^2 = a^2.
    Show that the equation of the tangent to the circle at (p,q) is px + qy = a^2

    3. A circle passes through the points C (4,4) and D (10,2).
    The line 2y = x + 4 is a tangent to the circle at C
    Find the euqation of the circle

    thanks
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  2. #2
    MHF Contributor Calculus26's Avatar
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    2. x^2 + y^2 = a^2


    x + ydy/dx = 0

    dy/dx = -x/y

    at (p,q) dy/dx = -p/q

    The tangent line is y = -p/q x + c

    at (p,q) q = -p^2/q +C

    q^2 + p^2 = Cq

    a^2/q = C

    y = -p/q x + a^2/q

    qy = -px +a^2

    result follows

    3.

    (x-h)^2 + (y-k)^2 = r^2

    2(x-h) + 2 (y-k)dy/dx = 0

    dy/dx = - (x-h)/(y-k)

    at (4,4) dy/dx = -(4-h)/(4-k)

    Since 2y = x + 4

    -(4-h)/(4-k) = 1/2

    2h -8 = 4 - k

    k = -2h + 4

    So

    (x-h)^2 + (y-k)^2 = r^2

    becomes

    (x-h)^2 + (y +2h -4) ^2 = r^2

    Use the points (4,4) and (10,2) in the above and you'll get an easy system of 2 eqs to solve for r and h and then k
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  3. #3
    MHF Contributor Calculus26's Avatar
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    make sure you use the edited version of my last post
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