JEE Main & Advanced Mathematics Quadratic Equations Question Bank Critical Thinking

  • question_answer 7) If \[a<b<c<d\], then the roots of the equation \[(x-a)(x-c)+2(x-b)(x-d)=0\] are [IIT 1984]

    A) Real and distinct

    B) Real and equal

    C) Imaginary

    D) None of these

    Correct Answer: A

    Solution :

    Given equation can be rewritten as \[3{{x}^{2}}-(a+c+2b+2d)x+(ac+2bd)=0\] Its discriminant D \[={{(a+c+2b+2d)}^{2}}-4.3(ac+2bd)\] \[={{\left\{ (a+2d)+(c+2b) \right\}}^{2}}-12(ac+2bd)\]      \[={{\left\{ (a+2d)-(c+2b) \right\}}^{2}}+4(a+2d)(c+2b)-12(ac+2bd)\]     \[={{\left\{ (a+2d)-(c+2b) \right\}}^{2}}-8ac+8ab+8dc-8bd\] \[={{\left\{ (a+2d)-(c+2b) \right\}}^{2}}+8(c-b)(d-a)\] which is +ve, since \[a<b<c<d\]. Hence roots are real and distinct.

adversite



LIMITED OFFER HURRY UP! OFFER AVAILABLE ON ALL MATERIAL TILL TODAY ONLY!

You need to login to perform this action.
You will be redirected in 3 sec spinner

Free
Videos