Question

If the radius of the circle passing through
the origin and touching the line x + y = 2
at (1,1) is r units, then the value of 3V2r
is
Solve this

Answer 1

Theorem: General equation of the cone with vertex at origin is homogenous of second degree of

the type ax2+by2+cz2+2hxy +2gzx +2fyz=0.

Cor.1: If

=

=

is a generator for the cone ax2+by2+cz2+2hxy +2gzx +2fyz=0 then prove that

D.R’s must satisfy eqn. of cone.

Proof: Given generator is If

=

=

= r

 any point on the generator is (lr, mr, nr), which lies on the cone

ax2+by2+cz2+2hxy +2gzx +2fyz=0---(1)

 This point (lr, mr, nr) must satisfy (1)

 we havea(lr)2+b(mr)2+c(nr)2+2h(mr)(nr)+2g(nr)(mr) +2f(mr)(lr)=0

i.e r2

(a(l)2+b(m)2+c(n)2+2hlm+2gln +2fmn)=0

But r2 ≠0,  al2+bm2+cn2+2hlm+2gln +2fmn =0

i.e D.R’s satisfy the eqn. of cone.

Cor.2: General equation of the cone with vertex at origin and passing through coordinate axis is

hxy +gzx +fyz=0.

Proof: Let General equation of the cone with vertex at origin be ax2+by2+cz2+2hxy +2gzx +2fyz=0.---(1)

If (1) passes through coordinate axis (they areas generators) then D.R.’s of x-axis, y-axis and z-axis

must satisfy eqn. (1) by cor.(1)

But D.R’s of x-axis are 1,0,0 , they satisfy eqn. (1) => a(1)1 + 0+0+0+0+0=0 , => a=0

Similarly D.R’s of y-axis are 0,1,0 and z-axis 0,0,1 must satisfy (1