Let S be the hyperboloid x 2 + y 2 = z2 + 1 and let P = (a, {3, 0) be a point on S in the (x, y )-plane. Show that there are precisely two lines through P entirely contained in S (Figure 18). Hint: Consider the line r(t) = (a +at, f3 + bt, t) through P. Show that r(t) is contained in S if (a, b) is one of the two points on the unit circle obtained by rotating (a, {3) through ± ~. This proves that a hyperboloid of one sheet is a doubly ruled surface, which means that it can be swept out by moving a line in space in two different ways.

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A: Here, given equation of line is,   And the given equation of plane is, From (1) and (2),

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A: Solution :-

Q: Determine a so that the line x-3/a=y+5/2=z+1/4 is parallel to the plane 2x+3y-5z=14.

A: To determine  so that the line  is parallel to the plane .

Q: the

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