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NDA 2 2024 Maths, question 47

Topic: Algebra

Passage · Q47 to Q48

The roots of the quadratic equation

a2(b2−c2)x2+b2(c2−a2)x+c2(a2−b2)=0a^{2}(b^{2}-c^{2})x^{2}+b^{2}(c^{2}-a^{2})x+c^{2}(a^{2}-b^{2})=0

are equal (a2≠b2≠c2)(a^{2}\ne b^{2}\ne c^{2}).

Q47

Which one of the following statements is correct?

  1. (a) a2,b2,c2a^{2},b^{2},c^{2} are in AP.
  2. (b) a2,b2,c2a^{2},b^{2},c^{2} are in GP.
  3. (c) a2,b2,c2a^{2},b^{2},c^{2} are in HP.Correct answer
  4. (d) a2,b2,c2a^{2},b^{2},c^{2} are neither in AP nor in GP nor in HP.

Solution.

Answer: (c) a2,b2,c2a^{2},b^{2},c^{2} are in HP.

  1. The coefficients sum to zero, so x=1 is a root.
  2. Equal roots therefore mean both roots are 1 and the linear coefficient equals minus twice the quadratic coefficient.
  3. Simplification gives b²c²+a²b²=2a²c², or 1/a²+1/c²=2/b².
  4. This is the HP condition for a²,b²,c²; the nondegenerate quadratic and equal-root condition rule out zero factors here.