Pure Mathematics

1. Given that the straight line y=c-3x does not intersect the curve xy=3, find the range of values for c.

2. Find the range of values for c for which the line y=cx+6 does not meet the curve 2x^2-xy=3.

3. Find the range of values for k for which 8y=x+2k is a tangent to the curve 2y^2=x+k

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  1. The slope of line cannot be that of curve or greater.

    slope line=-3

    slope curve>slope line

    -3<-3/x^2
    x^2>1
    x>1
    check x=10 slope line=-3
    slope curve=-3/100=-.03
    check x=1/2 slope curve=-12
    do the others the same way

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  2. number one is supposed to be -6<c<6
    number two is supposed to be c>5
    and number three is supposed to be k=8

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  3. the slope of xy=3 is y' = -3/x^2
    So, the slope is -3 at x=1

    The tangent to the curve at x=1 is

    y-3 = -3(x-1)
    y = -3x+6
    y = 6-3x

    so, for y = c-3x, if c < 6, the line falls below the curve. Similarly, for x<0, the line lies above the other branch of the curve if c > -6.

    So, -6 < c < 6

    Follow this logic for the other parts

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    posted by Steve

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