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If p(q-r)x^2+q(r-p)x+r(p-q)=0 has equal ...

If `p(q-r)x^2+q(r-p)x+r(p-q)=0` has equal roots, then prove that `2/q=1/p+1/rdot`

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If the quadratic equation x^(2)-2x-m=0 and p(q-r)x^(2)-q(r-p)x+r(p-q)=0 have common root such that second equation has equal roots. Then the value of m is

If each pair ofthe following three equations x: x^(2)+p_(1)x+q_(1)=0.x^(2)+p_(2)x+q_(2)=0x^(2)+p_(3)x+q_(3)=0, has exactly one root common, prove that (p_(1)+p_(2)+p_(3))^(2)=4(p_(1)p_(2)+p_(2)p_(3)+p_(3)p_(1)-q_(1)-q_(2)-q_(3)]

Knowledge Check

  • If roots of the equation (q-r)x^(2)+(r-p)x+(p-q)=0 are equal, then p,q and r are in

    A
    AP
    B
    GP
    C
    HP
    D
    AGP

  • If the equadratic equation 4x ^(2) -2x -m =0 and 4p (q-r) x ^(2) -2p (r-p) x+r (p-q)-=0 have a common root such that second equation has equal roots then the vlaue of m will be :

    A
    0
    B
    1
    C
    2
    D
    3

  • If p, q, r each are positive rational number such tlaht p gt q gt r and the quadratic equation (p + q - 2r)x^(2) + (q + r- 2p)x + (r + p - 2q) = 0 has a root in (-1 , 0) then which of the following statement hold good? (A) (r + p)/(q) lt 2 (B) Both roots of given quadratic are rational (C) The equation px^(2) + 2qx + r = 0 has real and distinct roots (D) The equation px^(2) + 2qx + r = 0 has no real roots

    A
    `(r + p)/(q) lt 2`
    B
    Both roots of given quadratic are rational
    C
    The equation `px^(2) + 2qx + r = 0` has real and distinct roots
    D
    The equation `px^(2) + 2qx + r = 0` has no real roots

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    If the roots of the equation (1-q+(p^(2))/(2))x^(2)+p(1+q)x+q(q-1)+(p^(2))/(2)=0 are equal then show that p^(2)=4q

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