Find the intersection of the spheres $ {{x}^{2}}+{{y}^{2}}+{{z}^{2}}+7x-2y-z=13 $ and $ {{x}^{2}}+{{y}^{2}}+{{z}^{2}}-3x+3y+4z=8 $
- $ x-y-z=1 $
- $ x-2y-z=1 $
- $ x-y-2z=1 $
- $ 2x-y-z=1 $
The Correct Option is D
Solution and Explanation
The correct option is(D): 2x−y−z=1.
given by x² + y² + z² + 7x - 2y - z - 13 = 0 and x² + y² + z² - 3x + 3y + 4z - 8 = 0. If these spheres intersect, then the equation S - S' = 0 represents the equation of their common intersection plane.
Therefore, by subtracting the second sphere equation from the first, we get:
(x² + y² + z² + 7x - 2y - z - 13) - (x² + y² + z² - 3x + 3y + 4z - 8) = 0
Simplifying further:
x² + y² + z² + 7x - 2y - z - 13 - x² - y² - z² + 3x - 3y - 4z + 8 = 0
This reduces to:
10x - 5y - 5z - 5 = 0
And finally:
2x - y - z = 1.
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Concepts Used:
Three Dimensional Geometry
Mathematically, Geometry is one of the most important topics. The concepts of Geometry are derived w.r.t. the planes. So, Geometry is divided into three major categories based on its dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.
Direction Cosines and Direction Ratios of Line:
Consider a line L that is passing through the three-dimensional plane. Now, x,y and z are the axes of the plane and α,β, and γ are the three angles the line makes with these axes. These are commonly known as the direction angles of the plane. So, appropriately, we can say that cosα, cosβ, and cosγ are the direction cosines of the given line L.
