Home
Class 12
MATHS
If x^2+a x+b=0a n dx^2+b x+c a=0(a!=b) h...

If `x^2+a x+b=0a n dx^2+b x+c a=0(a!=b)` have a common root, then prove that their other roots satisfy the equation `x^2+c x+a b=0.`

Text Solution

Verified by Experts

The correct Answer is:
`(bc_(1) - ab_(1)) (cb_(1) - ba_(1)) = aa_(1) - cc_(1))^(2)`
Subtracting the given equations, we get
` (a - b) x + c (b - c) = 0 rArr x = c ` is the common root
Thus, roots of `x^(2) + ax + bc = 0` are b and c and those of `x^(2) + bx + ca = 0`
are c and a . Also,` a + b = -c`. Thus, the required equation is
`x^(2) - (a + b) x + ab = 0`
`rArr x^(2) + cx + ab = 0` .
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • THEORY OF EQUATIONS

    CENGAGE PUBLICATION|Exercise CONCEPT APPLICATION EXERCISE 2.11|8 Videos

  • THEORY OF EQUATIONS

    CENGAGE PUBLICATION|Exercise CONCEPT APPLICATION EXERCISE 2.12|11 Videos

  • THEORY OF EQUATIONS

    CENGAGE PUBLICATION|Exercise CONCEPT APPLICATION EXERCISE 2.9|12 Videos

  • STRAIGHT LINES

    CENGAGE PUBLICATION|Exercise ARCHIVES (NUMERICAL VALUE TYPE)|1 Videos

  • THREE DIMENSIONAL GEOMETRY

    CENGAGE PUBLICATION|Exercise All Questions|291 Videos

Similar Questions

Explore conceptually related problems

If the two equations x^2+ax+b=0 and x^2+bx+a=0 ( aneb ) have a common root, show that the other roots are the roots of the equation x^2+x+ab=0 .

If x^2+ax+bc =0 and x^2+bx+ac =0 have a common root, show their other root satisfies the equation x^2+cx+ab =0

Knowledge Check

  • If x^2+mx+1 =0 and (b-c) x^2+(c-a)x+(a-b) =0 have both roots common,then

    A
    m=-2
    B
    m=-1
    C
    a,b,c are in AP
    D
    a,b,c are HP

    • Similar Questions

    Explore conceptually related problems

    Prove that , if the equations x^2+bx+ca=0 and x^2+cx+ab=0 have only non-zero common root then their other roots satisfy the equation t^2+at+bc=0 .

    If the equation x^2+ax+bc=0" and " x^2+bx+ca=0 have a common root, then a+b+c =

    If the equation x^2-3p x+2q=0a n dx^2-3a x+2b=0 have a common roots and the other roots of the second equation is the reciprocal of the other roots of the first, then (2q-2b)^2 . a. 36p a(q-b)^2 b. 18p a(q-b)^2 c. 36b q(p-a)^2 d. 18b q(p-a)^2

    If the equations x^2 + 2x + 3 = 0 and ax^2 + bx + + c = 0 have common root, then a : b : c is _____

    If the equations x^2+ax+b=0 and x^2+bx+a=0 have one common root. Then find the numerical value of a+b.

    If a and c are odd prime numbers and a x^2+b x+c=0 has rational roots , where b in I , prove that one root of the equation will be independent of a ,b ,c.

    If the equation x^(2)+2x+3=0andax^(2)+bx+c=0 , a,b,c in RR, have a cmmon root , then a : b : c is -

    Home
    Profile