Answer to Question #91550 in Analytic Geometry for Ra

Question #91550
Find the equation of the cyclinder with base x²+y²+z²+3x+3y-z=1, x+y+2z=2.
1
Expert's answer
2019-07-16T08:47:55-0400

Given plane equation is (x+y+2z-2)=0, sphere equation is (x²+y²+z²+3x+3y-z)=1


x2+232x+9494+y2+232y+9494+x^2+2\cdot{3 \over 2}x+{9 \over 4}-{9 \over 4}+y^2+2\cdot{3 \over 2}y+{9 \over 4}-{9 \over 4}+


+z2212z+1414=1+z^2-2\cdot{1 \over 2}z+{1 \over 4}-{1 \over 4}=1

(x+32)2+(y+32)2+(z12)2=234\big(x+{3 \over 2}\big)^2+\big(y+{3 \over 2}\big)^2+\big(z-{1 \over 2}\big)^2={23 \over 4}

Centre(u,v,w)=(32,32,12)Centre (-u,-v,-w)=\big(-{3 \over 2},-{3 \over 2},{1 \over 2}\big)


Radius  r=234=232Radius\ \ r=\sqrt{{23 \over 4}}={\sqrt{23} \over 2}


Length of the perpendicular from (32,32,12)\big(-\dfrac{3}{2},-\dfrac{3}{2},\dfrac{1}{2}\big) to the plane (x+y+2z-2) =0 is 


OM=ax1+by1+cz1+da2+b2+c2OM=\dfrac{\big|ax_1+by_1+cz_1+d\big|}{\sqrt{a^2+b^2+c^2}}

Here, a=1,b=1,c=2,d=2,x1=32,y1=32,z1=12a=1,b=1,c=2,d=-2,x_1=-\dfrac{3}{2},y_1=-\dfrac{3}{2},z_1=\dfrac{1}{2}



OM=3232+212212+12+22=263OM=\dfrac{\big|-\dfrac{3}{2}-\dfrac{3}{2}+2\cdot\dfrac{1}{2}-2\big|}{\sqrt{1^2+1^2+2^2}}=\dfrac{2\sqrt6}{3}


AM2=OA2OM2=(232)2(263)2=3712AM^2=OA^2-OM^2=({\sqrt{23} \over 2})^2-({2\sqrt6 \over 3})^2={37 \over 12}

The equation of the axis of the cylinder is 


xαl=yβm=zγn{x-\alpha \over l}={y-\beta \over m}={z-\gamma \over n}

Here, α=32,β=32,γ=12,l=1,m=1,n=2.\alpha=-\dfrac{3}{2},\beta=-\dfrac{3}{2},\gamma=\dfrac{1}{2}, l=1,m=1,n=2.


x+321=y+321=z122{x+\dfrac{3}{2} \over 1}={y+\dfrac{3}{2} \over 1}={z-\dfrac{1}{2} \over 2}

l2+m2+n2=6l^2+m^2+n^2=6

Radius  r=AM=3712=1116Radius\ \ r^*=AM=\sqrt{{37 \over 12}}={\sqrt{111} \over 6}

The equation of Right Circular Cylinder is, 


[n(yβ)m(zγ)]2+[l(zγ)n(xα)]2+[n(y-\beta)-m(z-\gamma)]^2+[l(z-\gamma)-n(x-\alpha)]^2++[m(xα)l(yβ)]2=(r)2(l2+m2+n2)+[m(x-\alpha)-l(y-\beta)]^2=(r^*)^2(l^2+m^2+n^2)

[2(y+32)1(z12)]2+[1(z12)2(x+32)]2+[2(y+{3 \over 2})-1(z-{1\over 2})]^2+[1(z-{1 \over 2})-2(x+{3\over 2})]^2++[1(x+32)1(y+32)]2=(1116)2(6)+[1(x+{3 \over 2})-1(y+{3 \over 2})]^2=({\sqrt{111} \over 6})^2(6)


4y2+z2+4944yz+14y7z+z2+4x2+4944y^2+z^2+{49 \over 4}-4yz+14y-7z+z^2+4x^2+{49 \over 4}-4xz7z+14x+x22xy+y2=372-4xz-7z+14x+x^2-2xy+y^2={37 \over 2}



5x2+5y2+2z22xy4xz4yz+14x+14y14z+6=05x^2+5y^2+2z^2-2xy-4xz-4yz+14x+14y-14z+6=0

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