Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is times the distance from P to the yz-plane.

Question

Find an equation for the surface consisting of all points P for which the distance from P to the x-axis is  times the distance from P to the yz-plane.

Quizplus · Accepted Answer

Let P have coordinates $(x,y,z)$. The distance from $P$ to the $x$-axis is $|y|$. The distance from $P$ to the $yz$-plane is $|x|$. Thus we have the condition $|y| = 11eaa3e5_54fe_e327_9f8f_cf1c2af8411a_TB5971_11 |x|$. Since $|y|$ is always non-negative, we can simply write $y = 11eaa3e5_54fe_e327_9f8f_cf1c2af8411a_TB5971_11 x$ or $y = -11eaa3e5_54fe_e327_9f8f_cf1c2af8411a_TB5971_11 x$. These are equations of planes passing through the origin, so their union is a double-sided plane passing through the origin. But we are only interested in the portion of this plane in which $|y| = 11eaa3e5_54fe_e327_9f8f_cf1c2af8411a_TB5971_11 |x|$. This segment lies in the first and fourth quadrants, so we only need to consider $y = 11eaa3e5_54fe_e327_9f8f_cf1c2af8411a_TB5971_11 x$. Substituting, we get $x^2 + y^2 = (1 + 11eaa3e5_54fe_e327_9f8f_cf1c2af8411a_TB5971_11^2)x^2 = 11eaa3e5_54ff_0a3a_9f8f_530d94877417_TB5971_11\cdot (x^2 + y^2 + z^2)$, which is the equation of a sphere centered at the origin with radius $r = \sqrt{\frac{1}{1 + 11eaa3e5_54fe_e327_9f8f_cf1c2af8411a_TB5971_11^2}} = \sqrt{\frac{1}{1 + (11eaa3e5_54fe_e327_9f8f_cf1c2af8411a_TB5971_11)^2}} = 	extbf{(C)}\ 11eaa3e5_54ff_0a3a_9f8f_530d94877417_TB5971_11$.

Find an Equation for the Surface Consisting of All Points