Question #141001
Find the equation of the cylinder whose base is the circle x^2 + y^2=4 , x+y+2z=3
1
Expert's answer
2020-11-03T16:41:14-0500

Given equation of base is

x2+y2=4x^2+y^2=4


so the center of the cylinder lies at (0,0,0) and radius=2


Since Its equation of axis is fiven by

x+y+2z=3x3+y3+2z3=1x+y+2z=3\\\to\dfrac{x}{3}+\dfrac{y}{3}+\dfrac{2z}{3}=1


therefore Dr's of the above line are (3,3,32)\dfrac{3}{2})


Let P(a,b,c) be the any point on the cylinder

Draw PM perpendicular to the axis of cylinder


Therefore PM=2


By distance formula,

AP2=(a0)2+(b0)2+(c0)2           (1)AP^2=(a-0)^2+(b-0)^2+(c-0)^2~~~~~~~~~~~`-(1)


Let MA be the projection of AP on the axis

MA=3(a0)+3(b0)+32(c0)32+32+(32)2\dfrac{3(a-0)+3(b-0)+\dfrac{3}{2}(c-0)}{\sqrt{3^2+3^2+(\frac{3}{2})^2}}


MA=3a+3b+32c9+9+94\to MA=\dfrac{3a+3b+\dfrac{3}{2}c}{\sqrt{9+9+\frac{9}{4}}}



MA=32(2a+2b+c)92\to MA=\dfrac{\frac{3}{2}(2a+2b+c)}{\dfrac{9}{2}}



MA=13(2a+2b+c)\to MA=\dfrac{1}{3}(2a+2b+c)



As AP2MA2=(radius)2^2-MA^2=(radius)^2 ,


a2+b2+c2)19(2a+2b+c)2=22a^2+b^2+c^2)-\dfrac{1}{9}(2a+2b+c)^2=2^2


9a2+9b2+9c24a24b2c28ab4bc4ac=365a2+5b2+8c28ab4bc4ac=36\to 9a^2+9b^2+9c^2-4a^2-4b^2-c^2-8ab-4bc-4ac=36\\\to 5a^2+5b^2+8c^2-8ab-4bc-4ac=36


Replacing a,b,c by x,y,z

Therefore the equation of the cylinder is

5x2+5y2+8z28xy4yz4xz=36.5x^2+5y^2+8z^2-8xy-4yz-4xz=36.


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