Question

Find the equation of the cylinder whose base is the circle x^2 + y^2 = 4, z = 0 and the axis is x= y/2 = z.

Answer 1

All sections of the cylinder by planes with equation z = z₀ are circles of radius 2 and center in the point of intersection of this plane and axis of the cylinder.

Equations of a circle with center (x₀, y₀) and radius r is

$$(x-x_0)^2+(y-y_0)^2=r^2$$

substituting x₀ = z, y₀ = 2z and r = 2 into this equation gives equation of the cylinder

$$(x-z)^2+(y-2z)^2=4$$ $$x^2-2xz+z^2+y^2-4yz+4z^2=4$$

$$x^2+y^2+5z^2-2xz-4yz-4=0$$

Answer: x² + y² +5z² - 2xz - 4yz - 4 = 0.