Question

If α, ß are roots of ax2 + bx+c=0 to so,
then the equation with roots α/ß ,β/α is:

​

Answer 1

ax

2

+bx+c=0

α+β=

a

−b

αβ=

a

c

(i) equation whose roots are

α

2

1

,

β

2

1

Sum of roots =

α

2

1

+

β

2

1

=

(αβ)

2

α

2

+β

2

=

(αβ)

2

(α+β)

2

−2αβ)

=

a

2

c

2

a

2

b

2

−

a

2c

=

c

2

b

2

−2ac

\

product =

α

2

1

β

2

1

=

c

2

a

2

∴ equation is

c

2

x

2

+(2ac−b

2

)x+a

2

=0

(ii)

aα+β

1

,

aβ+α

1

Sum ⇒

a

2

αβ+α

2

a+aβ

2

+αβ

aβ+aα+β

⇒

(a

2

+1)(αβ)+a(αβ)

(α+β)(1+a)

⇒

(a

2

+1)

a

c

+a(

a

2

b

2

−

a

2c

)

(1+a)(

a

b

)

⇒

(a

2

+1)c+a(b

2

−2ac)

−(1+a)b

⇒

c(1+a

2

)+ab

2

−(1+a)b

Product

c(1−a

2

)+ab

2

a

∴(c(1−a

2

)+ab

2

)x

2

+(1+a)bx+a=0

(iii) α+

β

1

,β+

α

1

Sum ⇒

β

αβ+1

+

α

αβ+1

⇒

αβ

α

2

β(αβ)+(α

2

+β

⇒

a

−b

+

c

−b

−b[

ac

a+c

]

Product

αβ

(αβ+1)

2

⇒

a

2

a

c

(c+a)

2

⇒(α+β)+

αβ

αβ

[

a+c

ac

]x

2

+bx+(a+c)=0