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Theory of Equations: Roots Equation Equation Whose Roots

JEE Maths question with a full step-by-step solution.

Question
If α\alpha and β\beta are the roots of the equation x2−2x−5=0x^2 - 2x - 5 = 0, then the equation whose roots are αα−1\dfrac{\alpha}{\alpha-1} and ββ−1\dfrac{\beta}{\beta-1} is
A6x2−12x+5=06x^2 - 12x + 5 = 0correct
B2x2+8x−5=02x^2 + 8x - 5 = 0
C−2x2+8x−5=0-2x^2 + 8x - 5 = 0
D6x2+12x−5=06x^2 + 12x - 5 = 0
Solution
Step 1: Let y=xx−1y = \dfrac{x}{x-1}, so x=yy−1x = \dfrac{y}{y-1}. Step 2: Substitute into x2−2x−5=0x^2 - 2x - 5 = 0 and multiply through by (y−1)2(y-1)^2:
y2−2y(y−1)−5(y−1)2=0y^2 - 2y(y-1) - 5(y-1)^2 = 0
y2−2y2+2y−5y2+10y−5=0  ⟹  −6y2+12y−5=0y^2 - 2y^2 + 2y - 5y^2 + 10y - 5 = 0 \implies -6y^2 + 12y - 5 = 0
6y2−12y+5=06y^2 - 12y + 5 = 0
 Correct answer: (1)
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