Search & Filtering
Identify and trace the conicoid 𝑦2 + 3𝑧2 = 𝑥. Describe its sections by the planes 𝑦 = 0 and 𝑧 = 0.
Find the projection of the line segment joining the points (1, −1, 6) and (4, 3, 2) on the line (𝑥−4)/3 = −𝑦 = 𝑧/5 .
Find the transformation of the equation 12𝑥2 − 2𝑦2 + 𝑧2 = 2𝑥𝑦 if the origin is kept fixed and the axes are rotated in such a way that the direction ratios of the new axes are 1, −3, 0; 3, 1, 0; 0, 0, 1
Examine which of the following conicoids are central and which are non-central. Also determine which of the central conicoids have centre at the origin.
(i) 𝑥2 + 𝑦2 + 𝑧2 + 4𝑥 + 3𝑦 − 𝑧 = 0
(ii) 2𝑥2 − 𝑦2 − 𝑧2 + 𝑥𝑦 + 𝑦𝑧 − 𝑧𝑥 = 1
(iii) 𝑥2 + 𝑦2 − 𝑧2 − 2𝑥𝑦 − 3𝑦𝑧 − 6𝑧𝑥 + 𝑥 − 2𝑦 + 5𝑧 + 4 = 0
Show that the conicoid 2𝑥2 + 2𝑦2 + 𝑥𝑦 − 𝑦𝑧 + 𝑧𝑥 + 2𝑥 − 𝑦 + 5𝑧 + 1 = 0 is
central. Hence find its centre.
Transform the equation 𝑥2 + 2𝑦2 − 6𝑧2 − 2𝑥 − 8𝑦 + 3 = 0 by shifting the
origin to (1, 2, 0) without changing the directions of the coordinate axes. What
object does this new equation represent? Give a rough sketch of it.
Show that the perpendiculars drawn from the origin to tangent planes to the cone 𝑥2 − 𝑦2 + 5z2 + 4𝑥𝑦 = 0 lie on the cone 𝑥2 − 𝑦2 + 𝑧2 + 4𝑥𝑦 = 0.
Find the equation of the cylinder with base 𝑥2 + 𝑦2 + 𝑧2 − 3𝑥 − 6𝑧 + 9 = 0, 𝑥 − 2𝑦 + 2𝑧 − 6 = 0.
Find the angle between the lines of intersection of the cone 4𝑥2 + 𝑦2 + 4𝑧2 + 4𝑦𝑧 + 2𝑧𝑥 = 0 and the plane 𝑥 + 2𝑦 + 3𝑧 = 0.
Find the equation of the sphere touching the plane 8𝑥 + 5𝑦 + 3𝑧 + 1 = 0 at (3, −1, −1) and cutting the sphere 𝑥2 + 𝑦2 + 𝑧2 − 2𝑥 + 𝑦 − 𝑧 − 6 = 0 orthogonally.
