Key Concepts and Formulas

• Equation of a Sphere: The general equation of a sphere is given by ${x^2} + {y^2} + {z^2} + 2ux + 2vy + 2wz + d = 0$.
• Intersection of Two Spheres: When two spheres intersect, their common points form a circle. This circle lies on a unique plane. The intersection of the spheres is precisely this circle.
• Plane of Intersection (Radical Plane): If the equations of two spheres are $S_1 = 0$ and $S_2 = 0$, where $S_1 = x^2+y^2+z^2+2u_1x+2v_1y+2w_1z+d_1$ and $S_2 = x^2+y^2+z^2+2u_2x+2v_2y+2w_2z+d_2$, then the equation of the plane containing their circle of intersection is given by: $S_1 - S_2 = 0$ This formula works because any point $(x,y,z)$ lying on the intersection of both spheres must satisfy both $S_1=0$ and $S_2=0$. Consequently, their difference $S_1-S_2$ must also be zero for these points. When we subtract the two sphere equations, the quadratic terms ($x^2, y^2, z^2$) cancel out, resulting in a linear equation in $x, y, z$. A linear equation in $x, y, z$ always represents a plane, and since this plane contains all points common to both spheres, it is the plane of their intersection.

Step-by-Step Solution

Step 1: Express the given sphere equations in the standard $S=0$ form. The given equations of the spheres are: Sphere 1: ${x^2} + {y^2} + {z^2} + 7x - 2y - z = 13$ Sphere 2: ${x^2} + {y^2} + {z^2} - 3x + 3y + 4z = 8$

To apply the formula $S_1 - S_2 = 0$, it is crucial to move all terms to the left-hand side of the equation, making the right-hand side zero. This ensures consistency with the general form $S=0$.

Equation of Sphere 1 (denoted as $S_1$): ${S_1}: {x^2} + {y^2} + {z^2} + 7x - 2y - z - 13 = 0$ Equation of Sphere 2 (denoted as $S_2$): ${S_2}: {x^2} + {y^2} + {z^2} - 3x + 3y + 4z - 8 = 0$

Step 2: Apply the formula $S_1 - S_2 = 0$ to find the plane of intersection. Now that both sphere equations are in the correct $S=0$ form, we can subtract $S_2$ from $S_1$. This operation is performed specifically to eliminate the $x^2$, $y^2$, and $z^2$ terms, transforming the combined equation into a linear form that represents a plane.

$S_1 - S_2 = 0$ $({x^2} + {y^2} + {z^2} + 7x - 2y - z - 13) - ({x^2} + {y^2} + {z^2} - 3x + 3y + 4z - 8) = 0$

Step 3: Perform the subtraction and simplify the resulting equation to find the plane. We carefully subtract each corresponding term. It is important to correctly distribute the negative sign to every term within the second parenthesis to avoid sign errors.

$(x^2 - x^2) + (y^2 - y^2) + (z^2 - z^2) + (7x - (-3x)) + (-2y - 3y) + (-z - 4z) + (-13 - (-8)) = 0$ $0 + 0 + 0 + (7x + 3x) + (-2y - 3y) + (-z - 4z) + (-13 + 8) = 0$ $10x - 5y - 5z - 5 = 0$

This is the equation of the plane containing the circle of intersection. To simplify it and present it in a more standard form (and to match the given options), we can divide all terms by the greatest common divisor of the coefficients, which is 5.

$\frac{10x}{5} - \frac{5y}{5} - \frac{5z}{5} - \frac{5}{5} = \frac{0}{5}$ $2x - y - z - 1 = 0$

Finally, moving the constant term to the right-hand side gives us the standard equation of the plane: $2x - y - z = 1$

This plane contains the circle formed by the intersection of the two given spheres. Therefore, the intersection of the spheres is identical to the intersection of one of the spheres (either $S_1$ or $S_2$) and this derived plane.

Common Mistakes & Tips

• Standard Form is Essential: Always ensure your sphere equations are in the form $x^2+y^2+z^2+2ux+2vy+2wz+d=0$ (i.e., all terms on the LHS, RHS is 0) before applying the $S_1-S_2=0$ formula. Forgetting to move the constant term can lead to incorrect results.
• Beware of Sign Errors: The most frequent mistake is mismanaging negative signs during subtraction. A good practice is to enclose the entire second sphere's equation in parentheses when subtracting, as shown in Step 2, to ensure the negative sign is distributed correctly to every term.
• Verification of Cancellation: The $x^2, y^2, z^2$ terms must cancel out. If they do not, it indicates an error in your subtraction or that the original equations were not standard sphere equations (e.g., coefficients of $x^2, y^2, z^2$ were not all 1 and equal).

Summary

To find the plane of intersection of two spheres, we first rewrite their equations in the standard form $S=0$. Then, we subtract the two equations ($S_1 - S_2 = 0$). This operation eliminates the quadratic terms, resulting in a linear equation that represents a plane. This plane is the radical plane, which contains the circle formed by the intersection of the two spheres. By simplifying the resulting linear equation, we obtain the desired plane equation.

The final answer is $\boxed{2x-y-z=1}$, which corresponds to option (A).