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Math Help - Quadratic Polynomials

  1. #1
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    Quadratic Polynomials

    Hi

    Could someone please help me get started on this question. I seriously have no clue how to do part (i).

    Given that y=mx + b is a tangent to the circle x^2 + y^2 = a^2.

    (i) Show that b^2 = a^2 (m^2 +1)

    (ii) If y = 3x + b and x^2 + y^2 = 25, find b.
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  2. #2
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    On the circle
    x^2+y^2=a^2

    differentiating:

    2xdx+2ydy=0

    So

    dy/dx=-x/y on the circle (slope of tangent at arbitary point).

    So if we call the point of intersection between the circle and the line x_0,y_0 then the tangent line has slope m=-x_0/y_0 and passes through the point x_0,y_0.

    That is:
    y_0=mx_0+b

    b=y_0-mx_0

    b^2=y_0^2+m^2x_0^2-2mx_0

    b^2=y_0^2+m^2x_0^2-2m(x_0/y_0)y_0^2

    b^2=y_0^2+m^2x_0^2+2m^2y_0^2

    b^2=y_0^2+m^2y_0^2+m^2(x_0^2+y_0^2)

    b^2=y_0^2+(x_0^2/y_0^2)y_0^2+m^2(x_0^2+y_0^2)

    b^2=y_0^2+x_0^2+m^2(x_0^2+y_0^2)

    b^2=a^2+m^2a^2

    b^2=a^2(1+m^2) as required
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