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If the system of equation ax + by + cz =...

If the system of equation ax + by + cz = 0, bx + cy + az = 0, cx + ay + bz = 0 has non-trivial solution then find the value of `|{:(bc-a^2,ca-b^2,ab-c^2),(ca-b^2, ab-c^2, bc-a^2),(ab-c^2, bc-a^2, ca-b^2):}|`

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det[[bc-a^(2),ca-b^(2),ab-c^(2)ca-b^(2),ab-c^(2),bc-a^(2)ab-c^(2),bc-a^(2),ca-b^(2)]]=det[[a,b,cb,c,ac,a,b]]^(2)

Prove that: |[bc-a^2,ca-b^2,ab-c^2],[ca-b^2,ab-c^2,bc-a^2],[ab-c^2,bc-a^2,ca-b^2]| is divisible by a+b+c and find the quotient.

Knowledge Check

  • Find the value of |{:(a,b,c),(a^(2),b^(2),c^(2)),(bc,ca,ab):}|

    A
    (a-b)(b-c)(c-a)(ab+bc+ca)
    B
    (a-b)(b-c)(c-a)(ab+bc+ca)abc
    C
    (a-b)(b-c)(c-a)(a+b+c)
    D
    None of these

  • if a gt b gt c and the system of equations ax + by + cz = 0, bx + cy + az 0 and cx + ay + bz = 0 has a non-trivial solution, then the quadratic equation ax^(2) + bx + c =0 has

    A
    at least one positive root
    B
    roots opposite in sign
    C
    positive roots
    D
    imaginary roots

  • If agtbgtc and the system of equations ax+by+cz=0, bx+cy+az=0, cx+ay+bz=0 has a non-trivial solution, then both the roots of the quadratic equation at^(2)+bt+c=0 are

    A
    non-real
    B
    of opposite sign
    C
    positive
    D
    complex

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    Read the following comprehension carefully, Let Delta ne 0 Delta^c denotes the determinant of cofactors , then Delta^c=Delta^(n-1) where n(gt 0) is the order of Delta . if a,b,c are the roots of the equation x^3-px^2+r=0 then the value of |{:(bc-a^2,ca-b^2,ab-c^2),(ca-b^2,ab-c^2,bc-a^2),(ab-c^2,bc-c^2,ca-b^2):}|

    If a+b+c=0 , then the value of (a^(2)+b^(2)+c^(2))/(bc+ca+ab) is

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