Question

please help me with this problem :
find the condition for the equations ax^2+bx+c=0 and a'x^2+b'x+c'=0 have reciprocal roots.

Answer 1

equations ax^2+bx+c=0 and a'x^2+b'x+c'=0 have reciprocal roots, if the product all roots must be equal 1.
prove :
c/a * a'/c' = c/a * a/c = 1
QED (^_*)

Answer 2

if p and q be the roots of ax^2+bx+c=0 then p+q = -b/a and p.q = c/a
as the roots are reciprocal then the roots of a'x^2+b'x+c'=0 will be 1/p and 1/q ??
hence 1/p + 1q = =b'/a' or (p+q)/pq = b'/a' and 1/p.1/q = c'/a' or 1/p.q = c'/a' ?

Answer 3

a'x^2+b'x+c'
a' and c' here means you have to replace a and c, so the equation becomes
cx^2 + bx + a = 0

Answer 4

then how do i find the condition that they have reciprocal roots as asked in the question?

Answer 5

like i said above, read my first coment

Answer 6

the answer given is b/b' = c/a'= a/c', i am not getting you
what is wrong in this process
if p and q be the roots of ax^2+bx+c=0 then p+q = -b/a and p.q = c/a as the roots are reciprocal then the roots of a'x^2+b'x+c'=0 will be 1/p and 1/q ?? hence 1/p + 1q = =b'/a' or (p+q)/pq = b'/a' and 1/p.1/q = c'/a' or 1/p.q = c'/a' ?

Answer 7

too long, but i see that. (btw, 1/p + 1/q = -b'/a')

Answer 8

the answer given is b/b' = c/a'= a/c' how does it come ?

Answer 9

let the roots of ax^2 + bx + c = 0 are p and q
we know that the sum of roots and the product of roots above is :
p+q = -b/a
p * q = c/a
respectively.

and supposed the roots of a'x^2 + b'x + c' = 0 are p' and q'
we know that the sum of roots and the product of roots above is :
p'+q' = -b'/a'
p' * q' = c'/a'

now will be proved that b/b' = c/a'= a/c'

known that p' = 1/p, and q' = 1/q

p'q' = 1/p * 1/q = 1/pq
c'/a' = 1/(c/a) = a/c or
=> cc' = aa'
=> c/a' = a/c'

p'+q' = 1/p + 1/q
-b'/a' = (p+q)/pq
-b'/a' = (-b/a)/(c/a) = -b/c
............................................... x (-1)
b'/a' = b/c or
=>b'c = ba'
=> c/a' = b/b'

therefore,
b/b' = c/a'= a/c'
QED

Answer 10

fantastic .. thanx a lot for your step by step explanation :)

Answer 11

YOU'RE WELCOME :)