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-y2z2 + xy + yz - zx = 1

(iii) x2 + y2 - z2 -2xy -3yz - 6zx + x - 2y + 5z + 4 = 0

Show that the conicoid 2x² + 2y² + xyyz + zx + 2xy + 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 + z²-3x6z + 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.

Show that the plane 2x + y + 2z = 0 is a tangent plane to the sphere x² + y² + z2-2x+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 + 2y5z + 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).

Prove that the length of the chord of a parabola which passes through the focus and which is inclined at 30° to the axis of the parabola is four times the length of the latus rectum.

Find the eccentricity, foci, centre and directrices of the ellipse  . Also 4 give a rough sketch of it.

Prove that the equation of a line through (x1, y1) and (x2, y2) can be expressed in the form

Show that  represents the equation of a line passing 1-4 7 1 through (2, 3) and (-4, 7).

Let P be the midpoint of the line segment joining the points A(a + b, b) and B(a - b, a + b). Find the slope of the line passing through P and Q (b,- a/2). Under what conditions on a and b, this line is parallel to the y-axis? 

Show that the line x = y touches the conic ax² + 2hxy + by² + 2gx + 2fy + c = 0, if f + g = 0.

Prove that the conic passing through the points of intersection of two rectangular hyperbolas is also a rectangular hyperbola.

Trace the conic x² 2xy + y² 3x + 2y + 3 = 0.