3D Geometry
Distance from plane to sphere
Grade 12
Question:
<p>The centre of a sphere is \(C = (-2, 1, +3)\). The distance of centre \(C\) from plane \(12x + 4y + 3z = 327\) and the radius of the sphere is \(\sqrt{4 + 1 + 4 + 155} = 13\). The shortest distance from the plane to the sphere is:</p>
<p>0</p>
<p>13</p>
<p>0 (they touch)</p>
<p>Cannot be determined</p>
Step-by-Step Solution
Key Concept: The shortest distance from a plane to a sphere equals the perpendicular distance from the sphere's center to the plane minus the radius. Use the point-to-plane distance formula: d = |ax₀ + by₀ + cz₀ + d|/√(a² + b² + c²).
Step 1: Find the perpendicular distance from center C = (-2, 1, 3) to plane 12x + 4y + 3z = 327. Using the point-to-plane distance formula: d = |12(-2) + 4(1) + 3(3) - 327|/√(12^2 + 4^2 + 3^2) d = |-24 + 4 + 9 - 327|/√(144 + 16 + 9) d = |-338|/√169 = 338/13 = 26 Step 2: The radius of the sphere is r = 13 (given). Step 3: The shortest distance from the plane to the sphere is: Shortest distance = (perpendicular distance from center) - (radius) Shortest distance = 26 - 13 = 13 ∴ Answer: C
Correct Answer: C