$x^3+y^3=8$ ,Number of straight lines through origin which do not meet this curve is

If a line, say, $y=mx$ does not meet the curve, then $x^3 + (mx)^3 = 8$ has no solutions in $x$.

For $m\neq -1$, we can divide by $1+m^3$ to get a solution. For $m=-1$, there is no solution.

The case of a vertical line through the origin, $x=0$ can easily be verified to intersect the curve.


All such lines with slope other than -1 meet the curve:

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Calculus

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