Question

If z = ax + by +cz +d = 0 then Independent variables is

Answer 1

Answer:

ANSWER

For given equation equation to posses non-trivial solution we must have,

∣

∣

∣

∣

∣

∣

∣

∣

a

1

1

1

b

1

1

1

c

∣

∣

∣

∣

∣

∣

∣

∣

=0

Now use the property of R

1

→R

1

−R

2

and R

2

→R

2

−R

3

∣

∣

∣

∣

∣

∣

∣

∣

a−1

0

1

1−b

b−1

1

0

1−c

c

∣

∣

∣

∣

∣

∣

∣

∣

=0

Now expand along first row

⇒(a−1)[(b−1)c−(1−c)]−(1−b)[0−(1−c)]=0

⇒(a−1)(b−1)c−(a−1)(1−c)+(1−b)(1−c)=0

⇒(1−a)(1−b)c+(1−a)(1−c)+(1−b)(1−c)=0

Divide both sides (1−a)(1−b)(1−c)

⇒

1−c

c

+

1−b

1

+

1−a

1

=0

⇒−

1−c

1−c−1

+

1−b

1

+

1−a

1

=0

⇒−1+

1−c

1

+

1−b

1

+

1−a

1

=0

∴

1−a

1

+

1−b

1

+

1−c

1

=1