Identify the type of quadric surface defined by the equation and find all x, y, and z intercepts of the resulting graph.

Question

16y²-25x²+z²=1 is the equation.

 

It wants: The quadratic surface is an _____ with x-intercept when x = _______, y-intercept when y = _______, z-intercept when z = ______.

 

I had plugged in that it was a hyperboloid with one sheet and for x = I said 1/5. Y = NONE, and Z = NONE. I believe x = 1/5 is the wrong answer, but I do not know how to get it. I understand that whichever one is negative is the axis, but I do not understand how to solve it. Thank you.

Answer 1

As you guessed, this is a one-sheet hyperboloid along the x-axis. The variable with a different sign or exponent for the general quadratic surface determines the orientation of the quadric surface. To find the intercepts of a specific axis, you need to make the other two variables equal to zero and solve the quadratic equation for the corresponding variable to the desired axis. Below are the specific calculations for this problem:

x-intercept: By plugging in y=0 and z=0 to 16y^2 - 25x^2 + z^2 = 1, we have x^2 = -1/25 < 0. Therefore, the answer is None.

y-intercept: By plugging in x=0 and z=0 to 16y^2 - 25x^2 + z^2 = 1, we have y^2 = 1/16. Therefore, the y-intercepts are (0, 1/4, 0) and (0, -1/4, 0).

z-intercept: By plugging in x=0 and y=0 to 16y^2 - 25x^2 + z^2 = 1, we have z^2 = 1. Therefore, the z-intercepts are (0, 0, 1) and (0, 0, -1).