Question
Find the equations of those tangent planes to the sphere x2 + y2 + z2 + 2x - 6z - 7 = 0 which intersect in the line 6x - 3y - 23 = 0, 3z + 2 = 0
Answer :
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The given sphere equation is: $$ x^2 + y^2 + z^2 + 2x - 6z - 7 = 0 $$ We can rewrite it in standard form by completing the squares: * For $x$: $x^2 + 2x = (x+1)^2 - 1$ * For $y$: $y^2$ (no change) * For $z$: $z^2 - 6z = (z-3)^2 - 9$ So the sphere becomes: $$ (x+1)^2 + y^2 + (z-3)^2 - 1 - 9 - 7 = 0 \implies (x+1)^2 + y^2 + (z-3)^2 = 17 $$ Hence, the sphere has center $C(-1, 0, 3)$ and radius $R = \sqrt{17}$. We are asked to find tangent planes that intersect in the line: $$ 6x - 3y - 23 = 0, \quad 3z + 2 = 0 \implies z = -\frac{2}{3} $$ Let the tangent plane be: $$ x X_0 + y Y_0 + z Z_0 + D = 0 $$ For a tangent plane to a sphere $(x+1)^2 + y^2 + (z-3)^2 = 17$ at point $(x_0, y_0,
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