Question

Show that the surfaces intersect at the given point,

z = 2xy², 8x² - 5y² - 8
z = -13, (1, 1, 2)

Answer 1

To show that the surfaces intersect at the given point (1, 1, 2), we need to substitute the given point into both equations and check if the left side of both equations equals the right side.

Step-by-step explanation:

To show that the surfaces intersect at the given point (1, 1, 2), we need to substitute the given point into both equations and check if the left side of both equations equals the right side. Let's start by substituting (1, 1, 2) into the first equation: z = 2xy². In this case, x = 1, y = 1, and z = 2. Substituting these values into the equation, we get 2 = 2*1*1², which simplifies to 2 = 2. Since this equation is true, the point satisfies the first equation.

Now let's substitute the same values into the second equation: 8x² - 5y² - 8z = -13. Substituting x = 1, y = 1, and z = 2 into the equation, we get 8*1² - 5*1² - 8*2 = -13. Simplifying, we get 8 - 5 - 16 = -13, which simplifies to -13 = -13. Since this equation is also true, the point satisfies the second equation.

Therefore, since the point (1, 1, 2) satisfies both equations, the surfaces intersect at this point.