Show that the plane 2π₯ + π¦ + 2π§ = 0 is a tangent plane to the sphere π₯ 2 + π¦ 2 + π§ 2 β 2π₯ + 2π¦ β 2π§ + 2 = 0.
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Expert's answer
2022-02-08T13:05:41-0500
Now we know, If the plane touches the sphere, then the perpendicular distance from the center of the sphere to the plane should be equal to the radius of the given sphere.
We have x2+y2+z2β2x+2yβ2z+2=0
On adding and subtracting terms and rearranging we get,
(xβ1)2+(y+1)2+(zβ1)2=1 using method of completing the squares.
So, the center of the sphere is (1,β1,1) and the radius is 1 since the general equation of a sphere is (xβg)2+(yβh)2+(zβI)2=r2 where r is the radius of the sphere.
Now, the distance Between (1,β1,1) and the plane 2x+y+2z=0 is;
βp=A2+B2+C2βAx+By+Cz+Dβ where Ax+By+Cz+D is the equation of the plane with x , y and z being the points through which the distance is to measured, which here is the center (1,β1,1).
βp=22+12+22β2β 1+1β (β1)+2β 1+0β=1
Therefore, the radius of the sphere and the distance 'p' between the center of the sphere and the plane are equal.
Hence, the plane is tangential to the sphere.
Now we know that equation of the line joining the center and the planeβs point of contact can be written in the cartesian form as
Axβx1ββ=Byβy1ββ=Czβz1ββ=Ξ»
where (x1β,y1β,z1β)=(1,β1,1) and A , B and C being the coefficients of x , y and z in the equation of the plane.
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