1. Sphere Def.1 Surface- Let f(x,y,z) be a polynomial of nth degree in x, y, z. Then the graph of the equation f(x,y,z)=0 defined as nth degree surface. Def 2. Reflection of a point in a plane- Let Be a given plane. Any point P not on will be on one of the two sides of. Def 3. Symmetry of a surface about a plane: If for each point P on a surface S, its symmetric about a plane also lies on S, then S is said to be symmetric about the plane and is said to be a plane of symmetry or a principal plane for S and the direction of the normal to is called a principal direction foe S.

2. Def 4. Symmetry about a point. If for each point on a surface S its symmetric point about M is also a point on S, then S is said to be symmetric about M and M is said to be symmetric for S. Example: Discuss all the symmetries for the folloeing surfaces: (1) (2) (3) (4) Notations: (i)By S the second degree expression (ii)By, the expression,where ( )is any point

3. (iii) by or the expression obtained by replacing (x, y, z) by ( ) in i.e Def 5. Sphere:- A sphere is the graph S all point of the 3-space which are at a fixed distance r > 0 from a fixed point C( ). The point C is called the centre and r is called the radius of the sphere. Standard form of Sphere:- To show that the equation of the sphere whose centre is origin and radius r is

4. General form:- To prove that the equation represent a sphere and radius. Example:- Find the equation of the sphere whose centre is (2,-3,4) and radius 5. Four point form:- To find the equation of the sphere though four given non – coplanar points. Let be the given (non-coplanar ) points. Example:- Find the equation of the sphere through the four point O(0,0,0), A(-a, b, c),B(a,- b, c), C(a, b,-c) and determine its radius.

5. Diameter form. To find the equation of the sphere on the join of as diameter. Example: Obtain the equation of the sphere described on the join the point A(1,2,3) and B(0,4,-1) as diameter. Exterior and Interior points of a sphere. Theorem1. If S= is the equation to a sphere, then the point P is the exterior or interior to the sphere according as Example: Are the following points exterior or interior to the sphere (i) (1,2,-1) (ii) (3,-2,1) (2,-2,1)

6. Sphere through a given circle:- The equation of any sphere through the circle of intersection of the sphere and the plane is where k is an arbitrary constant. Example: Find the equation of the sphere through the circle and the point (1,2,3) Def: Power of a point w. r. t a sphere:- If through a given point A is drawn any chord PP’ of the sphere, then AP,AP’ is called the power of A w. r. t the sphere

7. Theorem: - If through a given point is drawn any chord PP’ of the sphere, then AP,AP’ is constant and equal or If then AP,AP’ is constant and is equal to Def:- Tangent Plane If a sphere and a plane have exactly one common point, then the plane is said to touch the given sphere at the common point or is said to be a tangent plane to the sphere at the point. The (unique )common point is called point of contact.

8. Tangent lines:- Let a plane pouch the sphere S at the point N. Then for every other point P of the plane, the line Cuts the sphere in the unique point N. The line is called the tangent line. or A straight line which cuts a sphere in a unique point N is called a tangent line to the sphere at the point N. Condition of tangency of a plane and a sphere. To find the condition that the plane should touch the sphere

9. Example: Find the equation of the tangent plane (-1,4,-2) to the sphere Example : The equation of the sphere which pass through the circle And touch the line Def: Pole and Polar If through a point A is drawn any chord BC of a sphere, and D is the harmonic conjugate of A w.r. t B and C, then the locus at D is a plane called the polar plane of A w. r. t the sphere. Also the point A is called the pole of this polar plane.

10. Polar plane: To find the equation of the polar of the polar plane of the point w.r.t the sphere Pole of Plane: To find the pole of the plane w. r. t to the sphere Def: Conjugate points: Two planes s.t. the polar plane of either passes through the other, are called conjugate points. Def: Conjugate planes: Two planes s. t. the pole of either lies on the other are called conjugate planes. If the polar plane of any point on a line (l) w. r. t a sphere passes through a line (l’), then the polar plane of any point (l’) passes through (l)

11. Polar lines: Two lines, such that the polar of every point on either passes through the other, are called polar lines. Angle of intersection of two sphere: The angle of intersection of two spheres at a common point of intersection is the angle between the tangent plane to them at that point. Theorem: If are the radii of the two spheres, d the distance between their centres, and their angles of intersection, then

12. Orthogonal spheres:- Two sphere are said to be orthogonal (or to cut orthogonally) if their tangent planes at a point of intersection are at right angles to each other. Theorem Condition of orthogonally:- To the find the condition that the sphere. May cut orthogonally. Families of spheres. We know that the general equation of a sphere is Where u,v, w and d can have any values with the only restriction

13. These four constant can be chosen so that the resulting sphere satisfies four given conditions. In other words, a sphere is in general, determined by four conditions, Below are some examples of forms which these conditions can take. (1)The sphere passes through a given point. (2)The sphere touches a given plane or a given line (3) the sphere has a given radius (4) the centre of the sphere lies on a given line or on a given plane. Radical Plane: the locus of a point whose powers w.r.t. Two non-concentric sphere are equal is a plane called the radical plane of the two spheres.

14. Theorem : To find the equation of the radical plane of the two non concentric spheres. Properties of the radical plane (1)The joining the centres of the two spheres is perpendicular to their radical plane. (2)If two sphere intersect, their radical plane is the plane of intersection, i.e. the circle of intersection lies in the radical plane. (3)If two sphere touch each other externally or internally, their radical plane is then common tangent plane.

15. Theorem If S=0 and S’=0 be two sphere, then the equation where is a parameter,represents a family of spheres such that any two members of the system have the same radical plane. Co-axial system of spheres:- A system of sphere, any two members of which have the same radical plane, is called a co-axial system of spheres. Example: Two spheres of radii cut orthogonally. Prove that the radius of their common circle is