An elementary treatise on solid geometry by Smith, Charles, 1844-1916

By Smith, Charles, 1844-1916

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To find the prints where a given straight line cuts of the second the surface represented by the general equation degree. THE TANGENT PLANE. 7 + p(a,/8,7)=0 ............ (i). J Since this is a quadratic equation, any straight meets line the surface in two points. Hence all straight lines which lie in any particular plane meet the surface in two points. So that, all plane sections of a surface of the second degree are conies. In what follows surfaces of the second degree will generally be called conicoids.

X- x-2y-z= cut ? Is the Is the point (l,-3, 1) in the 1 Ans. cos~ |-v/2, acute, obtuse. Ex. 3. Find the equation of the plane through (1, 4, 3) perpendicular to the line of intersection of the planes 3x + 4y + 7z + 4 0, and + 2z+3 = also of the plane through (3, 1, - 1) perpendicular to the line of intersection of the planes 3x + y-z = 0, 5x-3y + 2z=Q. = Ans. 15x + y-7z + 2=Q. Ex. 4. Shew that the line +iiz+p = Q if IX ^= \ y - -- fj. + m/jL + nv =0, v is parallel to x-y Ans. , x+ lly + 14je=0.

35 II Determine the locus of a point which moves from two given straight lines. so as always to be equally distant Through two straight lines given in space two planes are at right angles to one another ; find the locus of their line of intersection. drawn A line of constant length has its extremities on 9. and shew 10. two given by it, find the equation of the surface generated that any point in the line describes an ellipse. straight lines ; Shew that the two straight + by + cz = 0, yz + zx + xy lines equations ax will represented by the be perpendicular if 1 - + 1 + I~ = n 0.

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