Question

The equations (b - C)x +(c - a)y+ (a - b) = 0
and (b³ - c³)x + (c³- a³)y + (a³ - b³) = 0 will
represent the same line if
(a) b + c = 0
(b) b = c, c = a and a = b or a + b + c = 0
(c) a + b = 0
(d) a + b + c #0​

Answer 1

Answer:

here is your answer

Step-by-step explanation:

The given equations are in the form ax+by+c=0 and px+qy+r=0 who represent the same line.

So,

p

a

=

q

b

=

r

c

=k when, k is a constant.

Here a=b

3

−c

3

,b=c

3

−a

3

,c=a

3

−b

3

and p=b−c,q=c−a,r=a−b

∴

b−c

b

3

−c

3

=k

⇒(b−c)(b

2

+c

2

+bc)=(b−c)k

⇒(b−c)(b

2

+c

2

+bc−k)=0

∴ either b−c=0

⇒b=c .......(i)

or (b

2

+c

2

+bc−k)=0

⇒b

2

+c

2

+bc=k ............(1)

Similarly c=a........(ii) and

c

2

+a

2

+ca=k ............(2)

Also a=b .......(iii) and

a

2

+b

2

+ab=k ............(3)

∴ from (i), (ii) and (iii), we have

a=b=c first condition

Again from (1) and (2), we have

b

2

+c

2

+bc=c

2

+a

2

+ca

⇒b

2

−a

2

=c(a−b)

⇒(b−a)(b+a)=c(a−b)

⇒b+a=−c

⇒a+b+c=0 second condition, we shall get the same considering (2) and (3).