Analytic Geometry Answers

Find the reciprocal cone of the cone x^2+z^2-2yz+4zx=0.

Find the new equation of the conicoid 2x^2+3y^2+5z^2-xy+z=1 when the coordinate system is transformed into a new system with the origin and with the coordinate axes having direction ratios 2,1,0; -1,2,5; 1,-2,1 with respect to the old system.

Find the point of intersection of the line x/4=y=z-1 and the plane 2x+y+z=5.
Also find the angle between them.

Identify the conic x^2+xy+2y^2-2x-5=0. Also trace it.

Check whether the following statements are true or false. Justify your answer with a short explanation or a counter example.
(i) Any line through the origin cuts the sphere x^2+y^2+z^2=4 at exactly two points.
(ii) The plane making intercept at the z-axis and parallel to the xy-plane intersects the cone x^2+y^2 = z^2(tan theta)^2 in a circle.
(iii) There exists no line with 1/under-root3 ,1/under-root2 ,1/under-root6 as direction cosines.
(iv) The tangent planes at the extremities of any axis of an ellipsoid are perpendicular.
(v) A section of an elliptic paraboloid by a plane is always an ellipse.
(vi) The curve xy^2+yx^2=0 is symmetric about the origin.
(vii) There exists a unique line which is perpendicular to the lines x=y=z/2 and x=y= -z.
(viii) The plane 3x+4y+2z=1 touches the conicoid 3x^2+2y^2=z^2=1.
(ix) The xy- plane intersects the sphere x^2+y^2+z^2+2x-z=2 in a great circle.
(x) Non degenerate conics are non-central.

the angle between vector p and vector q is cos inverse (-3b/2a),if |p|=|vector a +2bvector|. find vector q . if|p|=|q|

Find the radius of the circular section of the sphere x2  y2  z2  49 by the plane
2x  3y  z  5 14  0

Vertices B and C of a ABC lie along the line
2 1 0
2 1 4
x  y  z 
  . Find the area of the triangle given
that A has Coordinates 1,1,2 and line segment BC has length 5 units

Find the image of the point 1, 6,3 in the line
1 2
1 2 3
x y  z 
  . Also find the equation of the line
joining the given point and its image.