Question #348285

Find the centres of the spheres. which touch at the the plane 4x+3y=47 at the point (8,5,4) and which touch the Sphere x² + y ² +z² = 1

1
Expert's answer
2022-06-06T14:15:55-0400

Find the equation of normal to the plane at (8,5,4)(8,5,4) :


x84=y53=z40=k,\frac{x-8}{4}=\frac{y-5}{3}=\frac{z-4}{0}=k,

where kk is some number.

Then


x=4k+8,y=3k+5,z=4x=4k+8, y=3k+5, z=4

Center of required sphere(s) is C(4k+8;3k+5;4).C(4k+8;3k+5;4).

Radius of required sphere(s) is


R=(4k+88)2+(3k+55)2+(44)2R=\sqrt{(4k+8-8)^2+(3k+5-5)^2+(4-4)^2}


=25k2=5k.=\sqrt{25k^2}=5|k|.

Sphere x2+y2+z2=1x^2+y^2+z^2=1 has center (0;0;0)(0;0;0) and radius 1.1.


We will have equation:


(4k+80)2+(3k+50)2+(40)2(4k+8-0)^2+(3k+5-0)^2+(4-0)^2

=(5k+1)2=(5|k|+1)^2

16k2+64k+64+9k2+30k+25+1616k^2+64k+64+9k^2+30k+25+16

=25k2+10k+1=25k^2+10|k|+1

94k10k=10494k-10|k|=-104

k0:k\ge0: 84k=10484k=-104 No solution

k<0:94k+10k=104k<0: 94k+10k=-104

k=1k=-1


x=4(1)+8=4,y=3(1)+5=2,z=4x=4(-1)+8=4, y=3(-1)+5=2, z=4

The center of the sphere is C(4,2,4).C(4, 2, 4).


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