Question

determine an equation for the curve that is the intersection of s1 and s2.S₁ defined by x² +4y² +z² =6, and S₂ defined by x² +4y²+z²−2x=5

Answer 1

To find the curve of intersection, subtract the equation of S₁ from S₂ to eliminate similar terms, yielding x = 1/2. Plug this value into S₁'s equation to get 4y² + z² = 6 - 1/4, which simplifies to 4y² + z² = 24/4.

Step-by-step explanation:

To determine an equation for the curve that is the intersection of S₁ and S₂, where S₁ is defined by x² + 4y² + z² = 6, and S₂ is defined by x² + 4y² + z² - 2x = 5, we can simplify and compare both equations. If we subtract the S₁ equation from the S₂ equation, we eliminate the y and z terms, which are the same in both equations.

By simplifying, we get:
x² + 4y² + z² - 2x - (x² + 4y² + z²) = 5 - 6,
which simplifies to
-2x = -1. Therefore, x = 1/2. The intersection curve is now defined by a constant x value of 1/2, and a set of (y, z) values that satisfy x² + 4y² + z² = 6 when x = 1/2, leading to 1/4 + 4y² + z² = 6, which simplifies further to 4y² + z² = 24/4.

Answer 2

The equation of the curve that represents the intersection of the two surfaces S₁: x² + 4y² + z² = 6 and S₂: x² + 4y² + z² - 2x = 5 is found by subtracting the second equation from the first, solving for x, and then substituting back to find corresponding y and z. The resulting equation is x = 1/2, 4y² + z² = 23/4.

Step-by-step explanation:

To determine an equation for the curve that is the intersection of S₁, defined by x² + 4y² + z² = 6, and S₂, defined by x² + 4y² + z² - 2x = 5, we observe that the terms quadratic in x, y, and z are the same in both equations.

Subtracting the second equation from the first, we get:
2x = 1
or
x = 1/2

Substituting x = 1/2 back into either S₁ or S₂ provides us with the equation of the curve. For example, substituting into S₁:

(1/2)² + 4y² + z² = 6
1/4 + 4y² + z² = 6
4y² + z² = 24/4 - 1/4
4y² + z² = 23/4

Therefore, the equation of the curve of intersection is:

x = 1/2, 4y² + z² = 23/4