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The quadratic equations $${x^2} - 6x + a = 0$$ and $${x^2} - cx + 6 = 0$$ have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is

JEE · Math · previous-year question

  1. A.1
  2. B.4
  3. C.3
  4. D.2correct

Answer

D. 2

Explanation

Let the roots of equation $${x^2} - 6x + a = 0$$ be $$\alpha $$ and $$4$$ $$\beta $$ and that of the equation $${x^2} - cx + 6 = 0$$ be $$\alpha $$ and $$3\beta .$$ Then $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha + 4\beta = 6;\,\,\,\,\,\,\,4\alpha \beta = a$$ and $$\,\,\,\,\,\,\,\,\,\,\,\,\,\alpha + 3\beta = c;\,\,\,\,\,\,\,3\alpha \beta = 6$$ $$ \Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a = 8$$ $$\therefore$$ The equation becomes $${x^2} - 6x + 8 = 0$$ $$ \Rightarrow \left( {x - 2} \right)\left( {x - 4} \right) = 0$$ $$ \Rightarrow $$ roots are $$2$$ and $$4$$ $$ \Rightarrow \alpha = 2,\beta = 1$$ $$\therefore$$ Common root is $$2.$$

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