Step 1: Understanding the Concept:
The angle of intersection of two spheres is defined as the angle between the tangent planes to the spheres at any point on their circle of intersection. This angle is equal to the angle between the radii of the two spheres drawn to that point. Two spheres are said to be orthogonal if this angle is \(90^\circ\) (\(\pi/2\) radians).
Step 2: Key Formula or Approach:
Let the two spheres be \(S_1=0\) and \(S_2=0\).
\(S_1: x^2+y^2+z^2+2u_1x+2v_1y+2w_1z+d_1=0\)
\(S_2: x^2+y^2+z^2+2u_2x+2v_2y+2w_2z+d_2=0\)
The angle of intersection \(\theta\) is given by the formula:
\[ \cos\theta = \frac{r_1^2 + r_2^2 - D^2}{2r_1r_2} \]
where \(r_1, r_2\) are the radii of the spheres and D is the distance between their centers.
The condition for orthogonality (\(\theta = 90^\circ \implies \cos\theta=0\)) simplifies to:
\[ 2u_1u_2 + 2v_1v_2 + 2w_1w_2 = d_1 + d_2 \]
Step 3: Detailed Explanation:
Let's find the coefficients for each sphere.
Sphere 1: \(x^2+y^2+z^2+6y+2z+8=0\)
\(2u_1=0 \implies u_1=0\)
\(2v_1=6 \implies v_1=3\)
\(2w_1=2 \implies w_1=1\)
\(d_1=8\)
Center \(C_1 = (0, -3, -1)\).
Radius \(r_1 = \sqrt{0^2+3^2+1^2-8} = \sqrt{9+1-8} = \sqrt{2}\).
Sphere 2: \(x^2+y^2+z^2+6x+8y+4z+20=0\)
\(2u_2=6 \implies u_2=3\)
\(2v_2=8 \implies v_2=4\)
\(2w_2=4 \implies w_2=2\)
\(d_2=20\)
Center \(C_2 = (-3, -4, -2)\).
Radius \(r_2 = \sqrt{3^2+4^2+2^2-20} = \sqrt{9+16+4-20} = \sqrt{29-20} = \sqrt{9} = 3\).
Check for orthogonality:
The condition is \(2u_1u_2 + 2v_1v_2 + 2w_1w_2 = d_1 + d_2\).
LHS: \(2(0)(3) + 2(3)(4) + 2(1)(2) = 0 + 24 + 4 = 28\).
RHS: \(d_1 + d_2 = 8 + 20 = 28\).
Since LHS = RHS, the spheres are orthogonal. The angle of intersection is \(90^\circ\) or \(\pi/2\) radians.
Alternative method using the cosine formula:
Distance between centers \(D^2 = (-3-0)^2 + (-4-(-3))^2 + (-2-(-1))^2 = (-3)^2 + (-1)^2 + (-1)^2 = 9+1+1 = 11\).
\[ \cos\theta = \frac{r_1^2 + r_2^2 - D^2}{2r_1r_2} = \frac{(\sqrt{2})^2 + 3^2 - 11}{2(\sqrt{2})(3)} = \frac{2 + 9 - 11}{6\sqrt{2}} = \frac{0}{6\sqrt{2}} = 0 \]
Since \(\cos\theta=0\), \(\theta = \frac{\pi}{2}\).
Step 4: Final Answer:
The angle of intersection is \(\frac{\pi}{2}\), which corresponds to option (A).