Question

If the equation of the sphere through the circle \( x^2 + y^2 + z^2 = 9; \; 2x + 3y + 4z = 5 \) and through the point \( (1,2,3) \) is \( 3(x^2 + y^2 + z^2) - 2x - 3y - 4z = C \), then the value of \( C \) is

• A. \(11\)
• B. \(22\)
• C. \(36\)
• D. \(41\)
• E. \(54\)

Hint:

Sphere through a circle is obtained using \(S + \lambda P = 0\).

Answer 1

The Correct Option is B

Concept: A sphere passing through a circle (intersection of a sphere and a plane) can be written as: \[ S + \lambda P = 0 \] where \(S=0\) is the sphere and \(P=0\) is the plane.

Step 1: Write given equations. Sphere: \[ S = x^2 + y^2 + z^2 - 9 = 0 \] Plane: \[ P = 2x + 3y + 4z - 5 = 0 \] General sphere through circle: \[ x^2 + y^2 + z^2 - 9 + \lambda(2x + 3y + 4z - 5) = 0 \]

Step 2: Expand. \[ x^2 + y^2 + z^2 + 2\lambda x + 3\lambda y + 4\lambda z -9 -5\lambda = 0 \]

Step 3: Use given point \( (1,2,3) \). Substitute: \[ 1 + 4 + 9 + 2\lambda(1) + 3\lambda(2) + 4\lambda(3) - 9 - 5\lambda = 0 \] \[ 14 + 2\lambda + 6\lambda + 12\lambda - 9 - 5\lambda = 0 \] \[ 5 + 15\lambda = 0 \Rightarrow \lambda = -\frac{1}{3} \]

Step 4: Substitute back. \[ x^2 + y^2 + z^2 - \frac{2}{3}x - y - \frac{4}{3}z - \frac{22}{3} = 0 \] Multiply by 3: \[ 3(x^2 + y^2 + z^2) - 2x - 3y - 4z = 22 \] Thus: \[ C = 22 \]