Analytic Geometry Answers

Can any conic have its focus lying on the corresponding directrix? Give reasons for
your answer.

Identify an axis of revolution and generating conic of the surface 4x^2+25y^2+4z^2=100. Does this conic also generate x^2/4 + y^2/25 + z^2/4=1?Give reasons for your answer.

A rotating liquid forms a surface in the form of a paraboloid. The surface is 2m
deep at the centre and 10m across. Obtain an equation of the surface.

Find the equation of the conic of which one focus lies at (2,1) one directrix is
x + y = 0 and it passes through (1,4) Also identify the conic and reduce the conic you obtained above to standard form.
Draw a rough sketch of the conic obtained above.

Every conicoid has a unique centre.
Is the statement true? Give reason for your answers either with a
short proof or a counter example

ax^2+by^2+cz^2=d represents a sphere with radius √(a^2+b^2+c^2-d) where
a,b,c,d are positive real numbers.
Is the statement true? Give reason for your answers either with a
short proof or a counterexample.

Given any two conics C1 and C2 , at least one element of R× R will belong to
C1 ∩C2.
Is the statement true? Give reason for your answers either with a
short proof or a counterexample.

Any curve in 2D that is symmetric about the origin is also symmetric with respect to both the axes
Is the statement true? Give reason for your answers either with a
short proof or a counterexample.

Every cylinder has a circular base.
Is the statement true? Give reasons for your answers, either with a
short proof or a counterexample.

Find the locus of the point, the absolute value of the difference of the distances of
which from the points (2,2) and (0,0) is 2 . Identify the curve represented by the
locus.