Find the vector and cartesian equations of the sphere whose centre is $(1, 2, 3)$ and which passes through the point $(5, 5, 3)$

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Solution

Radius $= \sqrt{{(5 - 1)}^{2} + {(5 - 2)}^{2} + {(3 - 3)}^{2}} = \sqrt{16 + 9} = \sqrt{25} = 5$
Here $a = 5$ and $\to c = \to i + 2 \to j + \to k$
$∴$ Vector equation of the sphere is $∣ ∣ \to r - \to c ∣ ∣ = a$
$[\to r - (\to i + 2 \to j + \to k)] = 5$
Cartesian Equation : Let $\to r = x \to i + y \to j + z \to k$
$∣ ∣ ∣ (x \to i + y \to j + z \to k) - (\to i + 2 \to j + \to k) ∣ ∣ ∣ = 5$
$∣ ∣ ∣ (x - 1) \to i + (y - 2) \to j + (z - 3) \to k ∣ ∣ ∣ = 5$
${(x - 1)}^{2} + {(y - 2)}^{2} + {(z - 3)}^{2} = 25$
$x^{2} + y^{2} + z^{2} - 2 x - 4 y - 6 z - 11 = 0$

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