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For a, b,c non-zero, real distinct, the ...

For `a`, `b`,`c` non-zero, real distinct, the equation, `(a^(2)+b^(2))x^(2)-2b(a+c)x+b^(2)+c^(2)=0` has non-zero real roots. One of these roots is also the root of the equation :

A

`(b^(2)-c^(2))x^(2)+2a(b-c)x-a^(2)=0`

B

`(b^(2)+c^(2))x^(2)-2a(b+c)x+a^(2)=0`

C

`a^(2)x^(2)+a(c-b)x-bc=0`

D

`a^(2)x^(2)-a(b-c)x+bc=0`

Text Solution

Verified by Experts

The correct Answer is:
C
`(c )` ` (a^(2)+b^(2))x^(2)-2b(a+c)x+b^(2)+c^(2)=0`
`D=4b^(2)(a+c)^(2)-4(a^(2)+b^(2))(b^(2)+c^(2))`
`=-4(b^(4)-2b^(2)ac+a^(2)c^(2))`
`=-4(b^(2)-ac)^(2)`
For real roots, `D ge 0`
`implies -4(b^(2)-ac)^(2) ge 0`
`implies b^(2)-ac=0`
`implies` Roots are real and equal.
`:.` Roots are `(2b(a+c))/(2(a^(2)+b^(2)))`
`=(b(a+c))/(a^(2)+ac)`
`=(b)/(a)`
This root satisfies option `(c )`
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