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Sphere

Equation of a Sphere when radius is given and center is at origin

The equation of a sphere with radius $r$ and centre at $(0, 0, 0)$ is given by $x^{2} + y^{2} + z^{2} = r^{2}$ .

Equation of a Sphere when radius and centre are given

The equation of a sphere with radius $r$ and centre at $(a, b, c)$ is given by: $(x - a)^{2} + (y - b)^{2} + (z - c)^{2} = r^{2}$

which can also be represented as

x^{2} + y^{2} + z^{2} - 2 a x - 2 b y - 2 c z + (a^{2} + b^{2} + c^{2} - r^{2}) = 0

General form of the equation of a Sphere

The general form of the equation of a sphere is given by:

$x^{2} + y^{2} + z^{2} + 2 u x + 2 v y + 2 w z + d = 0$ where $u = - a$ , $v = - b$ , $w = - c$ and $d = α^{2} + b^{2} + c^{2} - r^{2}$

Equation of a Sphere when extreme points of the diameter are given

The equation of a sphere with $(x_{1}, y_{1}, z_{1})$ and $(x_{2}, y_{2}, z_{2})$ at the extremes of a diameter is given by:

(x - x_{1}) (x - x_{2}) + (y - y_{1}) (y - y_{2}) + (z - z_{1}) (z - z_{2}) = 0

Equation of a Sphere passing through four non-coplanar points

The equation of a sphere passing through four non-coplanar points is given by:

| \begin{matrix} x^{2} + y^{2} + z^{2} & x & y & z & 1 \\ x_{1}^{2} + y_{1}^{2} + z_{1}^{2} & x_{1} & y_{1} & z_{1} & 1 \\ x_{2}^{2} + y_{2}^{2} + z_{2}^{2} & x_{2} & y_{2} & z_{2} & 1 \\ x_{3}^{2} + y_{3}^{2} + z_{3}^{2} & x_{3} & y_{3} & z_{3} & 1 \\ x_{4}^{2} + y_{4}^{2} + z_{4}^{2} & x_{4} & y_{4} & z_{4} & 1 \end{matrix} | = 0

Condition for a plane to touch a Sphere

The condition that the plane $l x + m y + n z = p$ touches the sphere $x^{2} + y^{2} + z^{2} + 2 u x + 2 v y + 2 w z + d = 0$ is given by:

(l u + m v + n w + p)^{2} = (l^{2} + m^{2} + n^{2}) (u^{2} + v^{2} + w^{2} - d)

Tangent Plane to a Sphere

The equation of the tangent plane to the sphere $x^{2} + y^{2} + z^{2} + 2 u x + 2 v y + 2 w z + d = 0$ at point $(x_{1}, y_{1}, z_{1})$ is given by:

x x_{1} + y y_{1} + z z_{1} + u (x + x_{1}) + v (y + y_{1}) + w (z + z_{1}) + d = 0

Plane of Contact

The locus of the point of contact of tangent plane to the sphere $x^{2} + y^{2} + z^{2} + 2 u x + 2 v y + 2 w z + d = 0$ which passes through $(α, β, γ)$ is a circle which lies in the plane given by:

α x + β y + γ z + u (x + α) + v (y + β) + w (z + γ) + d = 0

This plane is known as the plane of contact or contact plane.

General equation of Sphere passing through the circle created by the interesection of another Sphere and a plane

The general equation of any sphere passing through the circle created by the intersection of the sphere $x^{2} + y^{2} + z^{2} + 2 u x + 2 v y + 2 w z + d = 0$ and the plane $A x + B y + C z + D = 0$ is given by:

$x^{2} + y^{2} + z^{2} + 2 u x + 2 v y + 2 w z + d + k (A x + B y + C z + D) = 0$ .

Great Circle

The plane section of a sphere is a circle and if the plane passes through the center of the sphere, then the section is called the great circle.

Angle between two Spheres

The angle between two spheres is calculated as: $\cos θ = \frac{r_{1}^{2} + r_{2}^{2} - d^{2}}{2 r_{1} r_{2}}$ , where $d$ is the distance between the spheres. It follows that for orthogonality, $d^{2} = r_{1}^{2} + r_{2}^{2}$ .

Condition for orthogonal intersection of two Spheres

The condition that two spheres $x^{2} + y^{2} + z^{2} + 2 u_{1} x + 2 v_{1} y + 2 w_{1} z + d_{1} = 0$ and $x^{2} + y^{2} + z^{2} + 2 u_{2} x + 2 v_{2} y + 2 w_{2} z + d_{2} = 0$ interesects orthogonally is given by:

2 (u_{1} u_{2} + v_{1} v_{2} + w_{1} w_{2}) = d_{1} + d_{2}

Length of tangent from a point to a Sphere

The length of the tangent drawn from $P (x_{1}, y_{1}, z_{1})$ to the sphere $x^{2} + y^{2} + z^{2} + 2 u x + 2 v y + 2 w z = 0$ is given by:

\sqrt{x_{1}^{2} + y_{1}^{2} + z_{1}^{2} + 2 u x_{1} + 2 v y_{1} + 2 w z_{1} + d}

PYQs

Sphere

1) Find the equation of the sphere in $x y z - p l a n e$ passing through the points $(0, 0, 0)$ , $(0, 1, - 1)$ , $(- 1, 2, 0)$ , $(1, 2, 3)$ .

[2018, 12M]

2) A plane passes through a fixed point $(a, b, c)$ and cuts the axes at the points $A$ , $B$ , $C$ respectively. Find the locus of the center of the sphere which passes through the origin $O$ and $A$ , $B$ , $C$ .

[2017, 15M]

3) Show that the plane $2 x - 2 y + z + 12 = 0$ touches the sphere $x^{2} + y^{2} + z^{2} - 2 x - 4 y + 2 z - 3 = 0$ . Find the point of contact.

[2017, 10M]

4) Find the equation of the sphere which passes though the circle $x^{2} + y^{2} = 4$ ; $z = 0$ and is cut by the plane $x + 2 y + 2 z = 0$ in a circle of radius 3.

[2016, 10M]

5) Find the positive value of $a$ for which the plane $a x - 2 y + z + 12 = 0$ touches the sphere $x^{2} + y^{2} + z^{2} - 2 x - 4 y + 2 z - 3 = 0$ . Also find the point of contact.

[2015, 10M]

6) Find the co-ordinates of the points on the sphere $x^{2} + y^{2} + z^{2} - 4 x + 2 y = 4$ the tangent planes at which are parallel to the plane $2 x - y + 2 z = 1$ .

[2014, 10M]

7) A sphere $S$ has points $(0, 1, 0), (3, - 5, 2)$ at opposite ends of a diameter. Find the equation of the sphere having the intersection of the sphere $S$ with the plane $5 x - 2 y + 4 z + 7 = 0$ as a great circle.

[2013, 10M]

8) Show that three mutually perpendicular tangent lines can be drawn to the sphere $x^{2} + y^{2} + z^{2} = r^{2}$ from any point on the sphere $2 (x^{2} + y^{2} + z^{2}) = 3 r^{2}$ .

[2013, 15M]

9) Find the points on the sphere $x^{2} + y^{2} + z^{2} = 4$ that are closest to and farthest from the point $(3, 1, - 1)$ .

[2011, 20M]

10) Show that the plane $x + y - 2 z = 3$ cuts the sphere $x^{2} + y^{2} + z^{2} - x + y = 2$ in a circle of radius 1 and find the equation of the sphere which has this circle as a great circle.

[2011, 12M]

11) Show that the equation of the sphere which touches the sphere $4 (x^{2} + y^{2} + z^{2}) + 10 x - 25 y - 2 z = 0$ at the point $(1, 2, - 2)$ and passes through the point $(- 1, 0, 0)$ is $x^{2} + y^{2} + z^{2} + 2 x - 6 y + 1 = 0$ .

[2010, 10M]

12) Show that every sphere through the circle $x^{2} + y^{2} - 2 a x + r^{2} = 0, z = 0$ cuts orthogonally every sphere through the circle $x^{2} + z^{2} = r^{2}, y = 0$ .

[2010, 20M]

13) Find the equation of the sphere having its centre on the plane $4 x - 5 y - z = 3$ , and passing through the circle

x^{2} + y^{2} + z^{2} - 12 x - 3 y + 4 z + 8 = 0;

3 x + 4 y - 5 z + 3 = 0

[2009, 12M]

14) Find the equations (in symmetric form) of the tangent line to the sphere $x^{2} + y^{2} + z^{2} + 5 x - 7 y + 2 z - 8 = 0$ , $3 x - 2 y + 4 z + 3 = 0$ at the point $(- 3, 5, 4)$ .

[2008, 12M]

15) A sphere $S$ has points $(0, 1, 0)$ , $(3, - 5, 2)$ at opposite ends of a diameter. Find the equation of the sphere having the intersection of the sphere $S$ with the plane $5 x - 2 y + 4 z + 7 = 0$ as a great circle.

[2008, 20M]

16) Find the equation of the sphere inscribed in the tetraahedron whosw faces are $x = 0$ , $y = 0$ , S$z=0 $a n d$ 3x+3y+6z=6$$.

[2007, 12M]

17) Show that the spheres $x^{2} + y^{2} + z^{2} - x + z - 2 = 0$ and $3 x^{2} + 3 y^{2} - 8 x - 10 y + 8 z + 14 = 0$ cut orthogonally. Find the center and radius of their common circle.

[2007, 15M]

18) Show that the locus of the centers of sphere of a co-axial system is a straight line.

[2005, 15M]

19) Find the equations of the tangent planes to the sphere $x^{2} + y^{2} + z^{2} - 4 x + 2 y - 6 z + 5 = 0$ , which are parallel to the plane $2 x + y - z = 4$ .

[2004, 12M]

20) A sphere of constant radius $r$ passes through the origin $O$ and cuts the co-ordinate axes at $A$ , $B$ and $C$ . Find the locus of the foot of the perpendicular from $O$ to the plane $A B C$ .

[2003, 15M]

21) Find the coordinates of the centre of the sphere inscribed in the tetrahedron formed by the planes

$x = 0$ , $y = 0$ , $z = 0$ and $x + y + z = a$ .

[2002, 12M]

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