Answer

Verified

445.8k+ views

**Hint:**A sphere with center \[\left( {a,b,c} \right)\] and radius r has the equation \[\left( {x - a} \right){\text{ }} + {\text{ }}\left( {y - b} \right){\text{ }} + {\text{ }}{\left( {z - c} \right)^2} = {r^2}\]

Or x2 + y2 +z2 + 2ux + 2vy + 2wz+d=0

Where center of sphere is y\[ = {\text{ }}\left( { - u, - v, - w} \right)\]

And radius of sphere is

= \[\sqrt {{u^2} + {v^2} + {w^2} - d} \]

**Complete step by step answer:**

Let

\[\vec r = x\vec i + j\vec y + z\vec k\,\], and \[r = \sqrt {{x^2} + {y^2} + {z^2}} \]

\[ = {r^2} = {x^2} + {y^2} + {z^2}\] [squaring both sides]

Now dot product of \[\vec r.(4\vec i + 2\vec j - 6\vec k)\]is:

\[ = 4\vec r.\vec i + 2\vec r.\vec j - 6\vec r.\vec k\]

\[ = 4x + 2y - 6z\]----(2) \[[\vec r.\vec i = x,\,\,\vec r.\vec j = y,\,\vec r.\vec k = z]\]

Our given equation is,

\[{\vec r^2} + \vec r.(4\vec i + 2\vec j - 6\vec k) - 11 = 0\]-----(3)

Using value of 1 and 2 in equation 3

\[{x^2} + {y^2} + {z^2} - (4x + 2y - 6z) - 11 = 0\]

\[{x^2} + {y^2} + {z^2} - 4x + 2y - 6z - 11 = 0\]

General equation of the sphere is

\[{x^2} + {y^2} + {z^2} + 2ux + 2vy{\text{ }} + 2wz + d{\text{ }} = 0\;\;\;\;\;\;\;\;\;\]----(5)

On comparing 4 with 5, we get;

\[2u = - 4,{\text{ }}2v = - 2,2w = 6\]and \[d = - 11\]

= \[u{\text{ }} - \] \[\dfrac{4}{2} = - 2,v = - \dfrac{2}{2} = 1,w = \dfrac{6}{2} = 3and\] \[d = - 1\]

\[u = - 2v = - 1,w = 3and\] \[d = - 11\]

Center \[ = \left( {{\text{ }} - u{\text{ }},v, - w} \right)\;\]\[ = \]\[\left( { - ( - 2)} \right), - \left( { - 1} \right), - 3\])

\[ = \left( {2,{\text{ }}1, - 3} \right)\]

And radius \[ = \] \[\sqrt {{u^2} + {v^2} + {w^2} - d} \]

Put \[u = {\text{ }} - 2,v = {\text{ }} - 1,{\text{ }}w = 3\]and \[d = {\text{ }} - 11\]

\[ = \sqrt {{{( - 2)}^2} + {{( - 1)}^2} + {{(3)}^2} + 11} \]

\[ = \sqrt {4 + 1 + 9 + 11} \]

\[ = \sqrt {14 + 11} \]

\[ = \sqrt {25} \]

\[ = 5\]

Hence, are the center and radius of sphere.

**Note:**a sphere is a three-dimension shape and it is mathematically defined as a set of points from the given point called “center” with an equal distance called radius “r” in the three-dimensional space of Euclidean space. The diameter “d’ is twice the radius. The pair of points that connect the opposite sides of a sphere is called “antipodes”. The sphere is sometimes interchangeably called “ball”.

The important properties of the sphere are:

A sphere is perfectly symmetrical.

It is not a polyhedron.

All the points on the surface are equidistant from the center.

It does not have a surface of centers.

It has constant mean curvature.

It has a constant width and circumference.

“while comparing the equation with the general equation we must take care of the signs”.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

At which age domestication of animals started A Neolithic class 11 social science CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

One cusec is equal to how many liters class 8 maths CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE