Find the equation of the sphere circumscribing the tetrahedron whose faces are $\frac{y}{b} + \frac{z}{c} = 0$ , $\frac{z}{c} + \frac{x}{a} = 0, \frac{x}{a} + \frac{y}{b} = 0$ and $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$

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

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x^{2} + y^{2} + z^{2} + (a^{2} + b^{2} + c^{2}) (\frac{x}{a} + \frac{y}{b} + \frac{z}{c}) = 0

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

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

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Solution

The correct option is A $x^{2} + y^{2} + z^{2} - (a^{2} + b^{2} + c^{2}) (\frac{x}{a} + \frac{y}{b} + \frac{z}{c}) = 0$
Solution:
$\frac{y}{b} + \frac{z}{c} = 0.......... (1)$
$\frac{z}{c} + \frac{x}{a} = 0.......... (2)$
$\frac{x}{a} + \frac{y}{b} = 0.......... (3)$
$\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1.......... (4)$
Four planes which are faces of tetrahedron are,
From (1), (2), and (3)
$y = b, z = c$
$\frac{z}{c} + \frac{x}{a} = 0$
$\Rightarrow \frac{c}{c} + \frac{x}{a} = 0$
$\Rightarrow x = - a$
$(x, y, z) = (- a, b, c) . . . . . . . . . . . (5)$
From (2), (3), and (4)
$x = a, z = c$
$\frac{x}{a} + \frac{y}{b} = 0$
$\Rightarrow \frac{a}{a} + \frac{y}{b} = 0$
$\Rightarrow y = - b$
$(x, y, z) = (a, - b, c) . . . . . . . . . . . (6)$
From (3), (4), and (1)
$x = a, y = b$
$\frac{y}{b} + \frac{z}{c} = 0$
$\Rightarrow \frac{b}{b} + \frac{y}{b} = 0$
$\Rightarrow z = - c$
$(x, y, z) = (a, b, - c) . . . . . . . . . . . (7)$
Now,
let the equation of sphere be,
$x^{2} + y^{2} + z^{2} + u x + v y + w z = 0$
putting the values from (5), (6), and (7) we get,
$a^{2} + b^{2} + c^{2} - u a + v b + w c = 0.............. (8)$
$a^{2} + b^{2} + c^{2} + u a - v b + w c = 0.............. (9)$
$a^{2} + b^{2} + c^{2} + u a + v b - w c = 0.............. (10)$
Solving equation (8), (9) , (10) we get,
$u = \frac{- a^{2} - b^{2} - c^{2}}{a}, v = \frac{- a^{2} - b^{2} - c^{2}}{b}, w = \frac{- a^{2} - b^{2} - c^{2}}{c}$
Putting these values in equation of sphere,
$\Rightarrow x^{2} + y^{2} + z^{2} + (\frac{- a^{2} - b^{2} - c^{2}}{a}) x + (\frac{- a^{2} - b^{2} - c^{2}}{b}) y + (\frac{- a^{2} - b^{2} - c^{2}}{c}) z = 0$
$\Rightarrow x^{2} + y^{2} + z^{2} + (- a^{2} - b^{2} - c^{2}) (\frac{x}{a} + \frac{y}{b} + \frac{z}{c}) = 0$
$\Rightarrow x^{2} + y^{2} + z^{2} - (a^{2} + b^{2} + c^{2}) (\frac{x}{a} + \frac{y}{b} + \frac{z}{c}) = 0$

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