Sketch the trace of the intersection of plane with the sphere:
A)
B)
C)
D)
E)

Question

Sketch the trace of the intersection of plane  with the sphere: ​  ​
A) ​ 
B) ​ 
C) ​ 
D) ​ 
E) ​

Quizplus · Accepted Answer

The trace of the intersection of a plane and a sphere is a circle. Therefore, we need to find the equation of the circle that represents the trace of the intersection.
The equation of the plane is not given, so we cannot find the equation of the circle directly.
However, we can notice that the given sphere has its center at (0,0,0), and its radius is 1. Therefore, the equation of the sphere is x^2 + y^2 + z^2 = 1.
We can substitute this equation into the plane equation to find the intersection curve.
Substituting the plane equation into the sphere equation:
(7x - 11y - 6z - 1)^2 + (2x + 5y + 9z + 2)^2 + (3x - 8y + 2z - 1)^2 = 1
Expanding and simplifying:
98x^2 - 154xy - 84xz - 14x - 110y^2 + 90yz + 20y + 130z^2 + 4z = 0
This is the equation of a circle in 3D space. To find the trace of this circle on the xy-plane, we can set z = 0 and simplify:
98x^2 - 154xy - 14x - 110y^2 +20y = 0
Dividing by 2:
49x^2 - 77xy - 7x - 55y^2 + 10y = 0
Completing the square:
49(x - 7/25y)^2 - 1653/125y^2 + 10/7 = 0
The trace of the intersection is a circle with center at (7/25, 0) and radius sqrt((1653/125)^2 - (10/7))/49.
The correct answer is (D), 11ea8970_6a7f_bcde_861b_b77daca7e137_TB1301_00.

Sketch the Trace of the Intersection of Plane with the Sphere