Given equation of base is
x2+y2=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=3→3x+3y+32z=1
therefore Dr's of the above line are (3,3,23)
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=(a−0)2+(b−0)2+(c−0)2 ‘−(1)
Let MA be the projection of AP on the axis
MA=32+32+(23)23(a−0)+3(b−0)+23(c−0)
→MA=9+9+493a+3b+23c
→MA=2923(2a+2b+c)
→MA=31(2a+2b+c)
As AP2−MA2=(radius)2 ,
a2+b2+c2)−91(2a+2b+c)2=22
→9a2+9b2+9c2−4a2−4b2−c2−8ab−4bc−4ac=36→5a2+5b2+8c2−8ab−4bc−4ac=36
Replacing a,b,c by x,y,z
Therefore the equation of the cylinder is
5x2+5y2+8z2−8xy−4yz−4xz=36.
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