A sphere is the locus of a point which moves in a space in such a way that its distance from a fixed point always remains constant.
`text(General Equation of the Sphere)`
`text(In Cartesian Form :)`
The equation of the sphere with centre (a, b, c) and radius r is
`(x – a)^2 + (y – b)^2 + (z – c)^2 = r^2 …….(i)`
In generally, we can write
`x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0`
Here, its centre is (-u, v, w) and radius =` sqrt(u^2 + v^2 + w^2 – d)`
`text(In Vector Form :)`
The vector equation of a sphere of radius a and Centre having position vector c is `|r – c| = a`
`text(Important Points to be Remembered :)`
(i) The general equation of second degree in x, y, z is `ax^2 + by^2 + cz^2 + 2hxy + 2kyz + 2lzx + 2ux + 2vy + 2wz + d = 0`
represents a sphere, if
(a) a = b = c (≠ 0)
(b) h = k = 1 = 0
The equation becomes
`ax^2 + ay^2 + az^2 + 2ux + 2vy + 2wz + d – 0 …(A)`
To find its centre and radius first we make the coefficients of `x^2, y^2` and `z^2` each unity by dividing throughout by a.
Thus, we have
`x^2+y^2+z^2 + (2u / a) x + (2v / a) y + (2w / a) z + d / a = 0 …..(B)`
∴ Centre is `(- u / a, – v / a, – w / a)`
and radius `= sqrt(u^2 / a^2 + v^2 / a^2 + w^2 / a^2 – d / a)`
`= sqrt(u^2 + v^2 + w^2 – ad) / |a| `.
(ii) Any sphere concentric with the sphere
`x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0`
is `x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + k = 0`
(iii) Since, `r^2 = u^2 + v^2 + w^2` — d, therefore, the Eq. (B) represents a real sphere, if `u^2 +v^2 + w^2 — d > 0`
(iv) The equation of a sphere on the line joining two points `(x_1, y_1, z_1)` and `(x_2, y_2, z_2)` as a diameter is
`(x – x_1) (x – x_2) + (y – y_1) (y – y_2) + (z – z_1) (z – z_2) = 0.`
A sphere is the locus of a point which moves in a space in such a way that its distance from a fixed point always remains constant.
`text(General Equation of the Sphere)`
`text(In Cartesian Form :)`
The equation of the sphere with centre (a, b, c) and radius r is
`(x – a)^2 + (y – b)^2 + (z – c)^2 = r^2 …….(i)`
In generally, we can write
`x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0`
Here, its centre is (-u, v, w) and radius =` sqrt(u^2 + v^2 + w^2 – d)`
`text(In Vector Form :)`
The vector equation of a sphere of radius a and Centre having position vector c is `|r – c| = a`
`text(Important Points to be Remembered :)`
(i) The general equation of second degree in x, y, z is `ax^2 + by^2 + cz^2 + 2hxy + 2kyz + 2lzx + 2ux + 2vy + 2wz + d = 0`
represents a sphere, if
(a) a = b = c (≠ 0)
(b) h = k = 1 = 0
The equation becomes
`ax^2 + ay^2 + az^2 + 2ux + 2vy + 2wz + d – 0 …(A)`
To find its centre and radius first we make the coefficients of `x^2, y^2` and `z^2` each unity by dividing throughout by a.
Thus, we have
`x^2+y^2+z^2 + (2u / a) x + (2v / a) y + (2w / a) z + d / a = 0 …..(B)`
∴ Centre is `(- u / a, – v / a, – w / a)`
and radius `= sqrt(u^2 / a^2 + v^2 / a^2 + w^2 / a^2 – d / a)`
`= sqrt(u^2 + v^2 + w^2 – ad) / |a| `.
(ii) Any sphere concentric with the sphere
`x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + d = 0`
is `x^2 + y^2 + z^2 + 2ux + 2vy + 2wz + k = 0`
(iii) Since, `r^2 = u^2 + v^2 + w^2` — d, therefore, the Eq. (B) represents a real sphere, if `u^2 +v^2 + w^2 — d > 0`
(iv) The equation of a sphere on the line joining two points `(x_1, y_1, z_1)` and `(x_2, y_2, z_2)` as a diameter is
`(x – x_1) (x – x_2) + (y – y_1) (y – y_2) + (z – z_1) (z – z_2) = 0.`
