(b) Transform the equation 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.
We are given the quadratic equation: \[ x^2 + 2y^2 - 6z^2 - 2x - 8y + 3 = 0 \] and we need to shift the origin to the point \( (1,2,0) \) without changing the coordinate axes. ### Step 1: Perform the Change of Variables Let: \[ X = x - 1, \quad Y = y - 2, \quad Z = z - 0 = z \] Rewriting \( x \) and \( y \) in terms of \( X \) ______ __________ ________ ______ ____ _________ _________ _________.
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Check whether the following statements are true or false. Justify your answer with a short explanation or a counter example
(i) The numbers are the direction cosines of a line.
(ii) The points and
are collinear.
(iii) The conic is degenerate
(iv) Intersection of the ellipsoid and the plan
is a circle. The conicoid
is non-central.
(vi) The line
(b) Find the equation of the plane which passes through the line of intersection of the planes and
and which is perpendicular to the plane
(a) 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)
(ii)
(iii)
(a) Let
(a) Identify and trace the conicoid Describe its sections by the planes
(c) Find the projection of the line segment joining the points (1, −1, 6) and (4, 3, 2) on the line
(c) Find the angle between the lines of intersection of the cone and the plane