Question #293614

Show that the plane 2π‘₯ + 𝑦 + 2𝑧 = 0 is a tangent plane to the sphere π‘₯ 2 + 𝑦 2 + 𝑧 2 βˆ’ 2π‘₯ + 2𝑦 βˆ’ 2𝑧 + 2 = 0.


1
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 π‘₯2+𝑦2+𝑧2βˆ’2π‘₯+2π‘¦βˆ’2𝑧+2=0\displaystyle π‘₯^2 + 𝑦^2 + 𝑧^2 βˆ’ 2π‘₯ + 2𝑦 βˆ’ 2𝑧 + 2 = 0

On adding and subtracting terms and rearranging we get,

(xβˆ’1)2+(y+1)2+(zβˆ’1)2=1\displaystyle (x-1)^2+(y+1)^2+(z-1)^2=1 using method of completing the squares.

So, the center of the sphere is (1, βˆ’1, 1)(1,\ -1,\ 1) and the radius is 11 since the general equation of a sphere is (xβˆ’g)2+(yβˆ’h)2+(zβˆ’I)2=r2(x-g)^2+(y-h)^2+(z-I)^2=r^2 where r is the radius of the sphere.

Now, the distance Between (1, βˆ’1, 1)(1,\ -1,\ 1) and the plane 2π‘₯+𝑦+2𝑧=0\displaystyle 2π‘₯ + 𝑦 + 2𝑧 = 0 is;

β‡’p=Ax+By+Cz+DA2+B2+C2\displaystyle \Rightarrow p=\frac{Ax+By+Cz+D}{\sqrt{A^2+B^2+C^2}} where Ax+By+Cz+DAx+By+Cz+D is the equation of the plane with xx , yy and zz being the points through which the distance is to measured, which here is the center (1, βˆ’1, 1)(1,\ -1,\ 1).

β‡’p=2β‹…1+1β‹…(βˆ’1)+2β‹…1+022+12+22=1\displaystyle \Rightarrow p=\frac{2\cdot1+1\cdot(-1)+2\cdot 1+0}{\sqrt{2^2+1^2+2^2}}=1

Therefore, the radius of the sphere and the distance 'pp' 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 plane’s point of contact can be written in the cartesian form as

xβˆ’x1A=yβˆ’y1B=zβˆ’z1C=Ξ»\displaystyle \frac{xβˆ’x_1}{A}=\frac{yβˆ’y_1}{B}=\frac{zβˆ’z_1}{C}=\lambda

where (x1 ,y1, z1)=(1, βˆ’1, 1)(x_1\ ,y_1,\ z_1)=(1,\ -1,\ 1) and AA , BB and CC being the coefficients of xx , yy and zz in the equation of the plane.

β‡’xβˆ’12=y+11=zβˆ’12=Ξ»β‡’x=2Ξ»+1, y=Ξ»βˆ’1, z=2Ξ»+1\displaystyle \Rightarrow\frac{x-1}{2}=\frac{y+1}{1}=\frac{z-1}{2}=\lambda\Rightarrow x=2\lambda+1,\ y=\lambda-1,\ z=2\lambda+1

Substituting these points in the equation of the plane we get:

0=2π‘₯+𝑦+2𝑧=2(2Ξ»+1)+(Ξ»βˆ’1)+2(2Ξ»+1)β‡’Ξ»=βˆ’13β‡’x=13, y=βˆ’43, z=13\displaystyle 0=2π‘₯ + 𝑦 + 2𝑧=2(2\lambda+1)+(\lambda-1)+2(2\lambda+1)\\ \Rightarrow \lambda=-\frac{1}{3}\Rightarrow x=\frac{1}{3},\ y=-\frac{4}{3},\ z=\frac{1}{3}

Hence the point of contact is;

(13, βˆ’43, 13)\displaystyle \left(\frac{1}{3},\ -\frac{4}{3},\ \frac{1}{3}\right)


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