# Q: The value of 'a' for which the equation (x-1) 2 =|x-a| has exactly three real solutions is/are A. 1/4 B. 3/4 C. 1 D. 5/4 Hi badiuddin the above question is from AIST FIITJEE there they gave the answer of this question as 3/4,1 and 5/4.....but they don't give the detail solution.....and u gave the ans only 5/4....

Q: The value of 'a' for which the equation (x-1)^{2}=|x-a| has exactly three real solutions is/are

A. 1/4

B. 3/4

C. 1

D. 5/4

Hi badiuddin the above question is from AIST FIITJEE there they gave the answer of this question as 3/4,1 and 5/4.....but they don't give the detail solution.....and u gave the ans only 5/4....

## 2 Answers

**Dear Tapasranjan**

**I have given the range of a for which given equation has real solution **

**3/4 ≤ a ≤ 5/4 **

**for exactly 3 solution ,**

**1) any equation has 1 root and the other equation has 2 root then 3 solution are possible .**

**2) Or both the equation has 2 roots , but one root in both the equation is common then also exactly 3 solution is possible .**

** 1st case is possible at end point of the calculated range of a ie a=3/4 and 5/4 (because at this point discrimnent of one equation is zero already derived in earlier post )**

**and for the second case**

**formed two equation x^{2} -3x+a+1 =0 for x≥a**

** x^{2} -x+1-a =0 for x<a **

**let β is a common root**

**then it is satisfied by both the equation **

** **

** **β** ^{2} -3**β

**+a+1 =0**

** **β** ^{2} -**β

**+1-a =0**

**solve **for β

**β =a **

**put this value of in any one of the equation **

**a^{2} -a+1-a =0**

**(a-1) ^{2} =0**

**a =1 **

**so for a=1 also exactly 3 solution exicts**

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