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Math Help - x-intercept of tangent to parabola

  1. #1
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    x-intercept of tangent to parabola

    Show that the tangent to the parabola Y=Ax^2 (for A does not equal 0) at the point where x = c will intersect the x axis at the point (c/2 , 0 . Where does it intersect the y axis.
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    Quote Originally Posted by frozenflames
    Show that the tangent to the parabola Y=Ax^2 (for A does not equal 0) at the point where x = c will intersect the x axis at the point (c/2 , 0 . Where does it intersect the y axis.
    To find the tangent find the derivative of y=ax^2 at x=c which is \lim_{h\rightarrow 0} \frac{a(x+h)^2-ax^2}{h}=2ac
    Thus, the slope of the tangent line is m=2ac. Now use the formula for finding equations of line by knowing thier slopes and the point that they pass through. That formula is y-y_0=m(x-x_0)this tangent line has slope m=2ac and passes through the point of tangentcy x=c thus y=ac^2 Thus, the equation of the tangent line is y-ac^2=2ac(x-c) thus, y=2acx-2ac^2+ac^2=2acx-ac^2 now set y=0 because this is the x-intercept thus 2acx-ac^2=0 solve for x thus x=c/2
    Q.E.D.
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    Where does it intersect the y Axis?
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    Quote Originally Posted by frozenflames
    Where does it intersect the y Axis?
    Easy since the equation of the tangent line is y=2acx-ac^2 the y-intersept is when x=0 thus, y=-ac^2
    Q.E.D.
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