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Find the ratios in which the spheres x^2+y^2+z^2=24 divides the line joining the points (-1,1,2) and (-4,4,8)

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    Only one sphere divides that line.

    The line goes through the origin (0,0,0). That is apprent by inspection. The sphere is centered at the origin. That is apparent from the equation of the sphere.

    Therefore the line (if extended) must intersect the sphere at two points equidistant from the origin, at radius sqrt(24)=4.899. The first point is inside the sphere; the second is outside; therefore the line is divided into two parts only.

    Find the point of intersection to complete the problem.

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