Analytic Geometry Answers

Check whether or not the conicoid represented by
5x^2+4y^2-4yz+2xz+2x-4y-8z+2=0 is central or not. If it is, transform the
equation by shifting the origin to the centre. Else, change any one coefficient to make the equation that of a central conicoid.

Q: x = y = z −1 does not intersect the cone x^2+y^2+z^2+2(xy+yz+zx)=0
Is the statement true? Give reason for your answer, either with a short proof or a counterexample.

Q: x + y + z = 0 touches the cone x^2+y^2+z^2+2(xy+yz+zx)=0
Is the statement true? Give reason for your answer, either with a short proof or a counterexample.

2x + 3y = 7z represents a line in three-dimensional space.
Is the statement true? Give reason for your answer, either with a short proof or a counterexample.

1/√2, 1/√3, 1/√5 form the direction cosines of a line.
Is the statement true? Give reason for your answer, either with a short proof or a counterexample.

Under a rotation of axes, a parabola can become a hyperbola.
Is the statement true? Give reason for your answer, either with a short proof or a counterexample.

Center of the circle (0,0) is tangent to the line x-3y =6

Find the equation of a locus of a point so that the square of its distance from (3,-3) is always numerically equal to the slope of the line joining it to the same point

Obtain the equation of the sphere having center on the line x/3= y/2= z/-5 and passing through the points
�(0,-2,-4) and (2,-1,-1)

Find the coordinates of the foot of the perpendicular from (-2,6) on the line
2x+3y-1=0. �