Analytic Geometry Answers

A plane passes through (a,b, c) and cuts the axes in A,B,C, respectively, where
none of these points lie on the origin O. Show that the centre of the sphere OABC
satisfies the equation a
x +
b
y +
c
z = 2.

Find the orthogonal canonical reduction of the quadratic form
−

Does there pass a plane through the lines (x+4)÷3 =y÷2 =(z−1)÷3 and x÷2 =(y−1)÷1 =(z+1)÷1?
Justify.

Check whether the points (1,−1,−2),(1,−4,2),(3,0,2),(4,−3−2) are coplanar
or not. If they are coplanar, write the equation of the plane they pass through.
Otherwise, change the coordinates of one of the points so that they become
coplanar. In this case, find the plane passing through them.

x^2 +y^2 +z^2 = xyz is the equation of a cone.Is it true or false?

If α,β, γ are direction ratios of a line, then so are α^2 ,β^2, γ^2. Is it true or false ?

Every planar section of an ellipsoid is an ellipse. Is it true or false ?

The circle (x−1)^2 +y^2 = 1,z = 0 lies inside the sphere centred at the origin, andhaving radius 2√2. Is it true or false ?

Show that the closed sphere with centre )7,3,2( and radius 10 in 3 R is contained in the
open cube P = {(x, y,z :) x − 2 <11, y − 3 <11, z − 7 <11}.