79. The plane x + y + z = 0 intersects the sphere x² + y² + z² = 9 along a circle. If (2, y, z) is a point on the circle, then the value of |y + z| is _____.    

The Plane x + y + z = 0 Intersects the Sphere x² + y² + z² = 9 Along a Circle. If (2, y, z) is a Point on the Circle, Then Find the Value of |y + z|

Correct Answer

Answer: 2

Detailed Solution

Since the point (2, y, z) lies on the circle formed by the intersection of the given plane and sphere, it must satisfy both equations simultaneously. The quickest way to begin is by substituting the known x-coordinate into the equation of the plane.

The equation of the plane is

x + y + z = 0

Substituting x = 2, we obtain

2 + y + z = 0

Therefore,

y + z = -2

The question asks for the absolute value of y + z.

Hence,

|y + z| = |-2| = 2

Thus, the required value is 2.

Verification Using the Sphere Equation

Although the answer is already obtained from the plane equation, it is useful to verify that such a point actually exists on the sphere.

The sphere is given by

x² + y² + z² = 9

Substituting x = 2,

4 + y² + z² = 9

or

y² + z² = 5

We have already found that

y + z = -2

Using the identity

(y + z)² = y² + z² + 2yz

we get

4 = 5 + 2yz

Therefore,

yz = -1/2

This confirms that real values of y and z exist, proving that the point indeed lies on the circle. Hence, the obtained answer is completely valid.

Why the Plane Equation Alone Solves the Problem

Many students immediately begin solving both equations together, making the problem unnecessarily lengthy. Since the required quantity is simply y + z, and the plane equation directly relates x, y, and z, substituting the known value of x immediately gives the required answer. The sphere equation only confirms the existence of the point and is not needed to compute |y + z|.

Concept Behind the Question

Whenever a point lies on the intersection of two geometric objects, it must satisfy both equations simultaneously. If the required quantity can be obtained directly from one equation, there is no need to perform additional calculations using the second equation. This approach saves valuable time during competitive examinations and improves accuracy.

Final Answer

|y + z| = 2

Answer: 2