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9-4 1 uv # ~ “ o ‘ il Zz E, , | 910.10. RIGHT CIRCULAR CONE, , Definition. A right circular cone is a surface generated by a line which moves, , »wch a way that it passes through a fixed point (called the vertex) and it makes a, 4 owant angle @ with a fixed straight line through the vertex., , The constant angle 9 is called the semi-vertical angle of the cone and the fixed, wight line through the vertex is called the axis of the cone., , AA) The section of right circular cone by a plane perpendicular to its, "818 a circle,, . i Abe the vertex, AO the axis and @ be the semi-vertical, “Be of the right circular cone. Now draw any plane, , ¥, Pendicular to the axis AO meeting it at the point O., , vy, j § <, ind maider any point P on the section of the cone by this plane, in P4 N, , Mich in ne OP isa line lying on the plane section of the cone, "engin rendicular to the axis AO of the cone. Hence OP is, , “Valed ‘ularto the axis AO and thus the triangle POAis a right, “angle with <PAQ = @,

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| Iknte QF III ie LAUT LW |, , , , , , , , , , , , , , trample 1, Find the equation of a right circular cone whose vertex is (0, 0, 0),, rans and semt-vertical angle ts a. [Bilaspur 1994; Ravishankar 90], , N, , § “lution. For the given cone, the, , , , , “'8(0, 0, 0), semi-vertical angle is, , Mt the dinane! . ‘ ., ‘rection cosines of its axis, 51.0.0, , P(x y 2), , , , : ex, "3. ¥,2) be any point on the, , "et direction ratios of its, ‘8 Line OP wil] bex-Oy-O ¢, , ‘, Z

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sat ROSIN NRT cm naan Vector Analysis and Geometry, , Also, OP makes an angle a with the axis of the cone, and therefore the equation ©, of the required cone 18, , , , , , , , x.l+y.0+2z.0 x, cos Ot = a, \x? + y? +2? Vx? + y? +2?, or (x? +y? + 2”) cos” a=x? or x? =(y? +27) cot” a, or y? +22 =x? tan” o.

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— —- valk UL,, , Example 2. Find the equation of the right circular co, origin, axts is z-axis and the semi-vertical angle is a., , [Ravishankar 1993; Bi, , z) be any point on the surface of, , nerating line of the cone and for, angle @ with OZ., Here direction ratios of the generating line OP are, , x-Qy-Qz-0 .e.,x, y, zand direction cosines of OZ are, 0,0, 1, , Solution. Let PCS: y., the cone, so that OP is a ge, all positions of P it make, , laspur 2005, 2015, 2017), , AZ, , x.0+ y.04+2.1, cos O = —___, , , , , , , , x? 4 y2 +22, => a 4.9 gh al peat, => x+y? 427 = 22 (14 tan? a), = x? + y? = 27 tan? g, , which is the equation of required cone., Example 3. Find the eauatian af tha wi-bs —: 1

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Thus is required equation of the right circular cone,, , fxample 5. Find the equation of a right circular cone whose vertex is (2, 1, , 5),, ss parallel to OY and the semi-vertical angle is 45°., , [Ravishankar 2009], solution. Since the axis of the required z, , “}** parallel to y-axis, the direction cosines of, ae ins of the cone will be [0, 1, O}., , 4a ‘et Plx, y, 2) be any point of the surface of, ‘”e, then the direction-ratios of its, “Wor AP will be x - 2 y — 1, z+ 3., , teqye . ~ 2).04+(y -1).14+ (2+ 3).0, , , , , , , , , We~ a + (y - 1)? (2+ 3), ae +(2 +3)? = W2(y- DI?, ' 2° ~ de 4 2y + 62 +:12= 0, , Sth, , By “atic, | bean “ niet required one. boas to ee oe sth vertex at, , r \¢, , a, , ., , , , y