Theory of EquationsmediumFree

Theory of Equations: Let Roots Equation Roots Equation

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

Question
Let α,β\alpha, \beta be the roots of the equation (x−a)(x−b)=c(x-a)(x-b) = c, c≠0c \neq 0. Then the roots of the equation (x−α)(x−β)+c=0(x-\alpha)(x-\beta) + c = 0 are
Aa,ba, bcorrect
Bb,cb, c
Ca,ca, c
Da+c,b+ca+c, b+c
Solution
Step 1: Since α,β\alpha, \beta are roots of (x−a)(x−b)−c=0(x-a)(x-b) - c = 0, the monic quadratic with roots α,β\alpha, \beta is:
(x−α)(x−β)=(x−a)(x−b)−c(x-\alpha)(x-\beta) = (x-a)(x-b) - c
 Step 2: Therefore:
(x−α)(x−β)+c=(x−a)(x−b)−c+c=(x−a)(x−b)(x-\alpha)(x-\beta) + c = (x-a)(x-b) - c + c = (x-a)(x-b)
 Step 3: The equation (x−α)(x−β)+c=0(x-\alpha)(x-\beta) + c = 0 becomes (x−a)(x−b)=0(x-a)(x-b) = 0, with roots x=ax = a and x=bx = b. Correct answer: (1)
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