stuck on math question

Question

The surface x^2 + y^2 = 7 is a circular cylinder.

What is an equation for the cross section at y = −7 using x and z.

What is an equation for the cross section at y = 0 using x and z.

What is an equation for the cross section at y = 7 using x and z.

How can I solve this?

Answer 1

You solve this by simply substituting the value for y.

Let's first do the second case, y = 0.

x² + 0² = 7

x = ±√7

The solution are two vertical lines, each described by two equations

x = +√7, y = 0

x = -√7, y = 0

Note that the variable z does not appear in the equation for the cylinder, because it is vertical:

A point (x, y, z) lies on the cylinder as longs as x²+y²=7, no matter what z is.

For the same reason the variable z does not appear in either solution -- the lines are vertical.

You can visualize the first solution as an intersection of two vertical planes,

("vertical" means parallel to the z-axis).

One plane intersects the x-axis at +√7, and since the equation does not contain y,

the plane is parallel to the y-axis.

The other plane intersects the y-axis in the origin and is parallel to the x-axis.

(It actually contains both the x-axis and the z-axis).

Your teacher seems to want a single equation using x and z.

That is not possible.

First a single equation in a 3D space describes a surface, not a curve.

Secondly the equations for the cross section will be independent of z, because so are the

surfaces forming the cross section.

If you apply the same approach to y=-7 and y=7 you will need to solve the equation

x² = -42

which does not have a solution in real numbers.

Geometrically viewed, the cylinder is centered at the origin, and has radius √7,

which is something less than 3.

So it has no intersection with the planes y=-7 and y=7.