Analytic Geometry Answers

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.

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.