Find the equation of the sphere passing through the points $(3, 0, 0), (0, - 1, 0)$ and $(0, 0, - 2)$ and having the centre of the plane $3 x + 2 y + 4 z = 1$ .

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Solution

Equation of sphere
$x^{2} + y^{2} + z^{2} + 2 u x + 2 v y + 2 w z + d = 0$
It passes through
$(3, 0, 0)$ $9 + 6 u + d = 0$ .....(i)
$(0, - 1, 0)$ $(1 - 2 v + d = 0$ ....(ii)
$(0, 0, - 2)$ $4 - 4 w + d = 0$ ....(iii)
Also centre $(- u, - v, - w)$ lies on the plane
$3 x + 2 y + 4 z = 1$
$- 3 u - 2 v - 4 w = 1$ .....(iv)
By solving equations (i), (ii), (iii), and (iv)
We get $u = - \frac{4}{3}$
$v = 0$
$w = \frac{3}{4}$
$d = - 1$
then putting the values of u, v, w, d in the above equation
$6 (x^{2} + y^{2} + z^{2}) - 16 x + 9 z - 6 = 0$ .

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