Question

A variable sphere passes through the points (0, 0, + c) and cuts thelines y = x tan.a, z=c; y=-x tan a, z=-c in the points P and P'.If PP has constant length 2a; show that the centre of the spherelies on the line z=0, x2 + y2 =(a2-c2) cosec^2 (2a )​

Answer 1

Answer:

ayz+bzx+cxy=xyz

Step-by-step explanation:

Correct option is

A

ayz+bzx+cxy=xyz

C

x

a

=

y

b

=

z

c

=1

Let the plane be

α

x

+

β

y

+

γ

z

=1.

It passes through (a,b,c)

∴

α

a

+

β

b

+

γ

c

=1. ...(1)

Now, coordinates of the points A,B,C are (α,0,0),(0,β,0) and (0,0,y) respectively.

Equation of the plane through A,B,C parallel to coordinate plane are

x=α ...(2)

y=β ...(3)

and z=γ. ...(4)

The locus of their point of intersection will be obtained by eliminating α,β,γ from these with the help of the relation (1).

We thus get

x

a

+

y

b

+

z

c

=1,

i.e.,ayz+bxz+cxy=xyz.