Math Answers

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.

Show that the plane 2𝑥 + 𝑦 + 2𝑧 = 0 is a tangent plane to the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 − 2𝑥 + 2𝑦 − 2𝑧 + 2 = 0.

Find the distance of the origin from the plane which passes through (2, 1, 8) , (1, 0, 2) and (−3, 4, 6)

Find the equation of the plane which passes through the line of intersection of the planes 3𝑥 + 4𝑦 − 5𝑧 = 9 and 2𝑥 + 6𝑦 + 6𝑧 = 7 and which is perpendicular to the plane 3𝑥 + 2𝑦 − 5𝑧 + 6 = 0

Find the equations of the line through (1,3, 4 ) and parallel to the line joining the points (−4, 5, 3) and (8, 9, 7)