Analytic Geometry Answers

The circle (x−1)

2 +y

2 = 1,z = 0 lies inside the sphere centred at the origin, and

having radius 2√

2.

Give the equation of a conic which is symmetric to the line x+2 = 0.

Find the equation of the hyperbola with vertices (1,−4) and (1,4), and foci at

(1,−6) and (1,6)

The normals at any point P of the ellipsoid x

2

9 +

y

2

4 +z

2 = 1 meet the coordinate

planes in Q1,Q2,Q3, respectively. Show that PQ1 : PQ2 : PQ3 :: 9 : 4 : 1.

Find the equation of the normal to the solid 2x

2 −y

2 +8z

2 = 11 at a point where it

intersects the line x−3 = z =

y+1

2

.

Find the equation of the normal to the solid 2x

2 −y

2 +8z

2 = 11 at a point where it

intersects the line x−3 = z =

y+1

2

.

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.

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

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