Let (x1, x2),(x2, x3) and (x3, x1) are respectively the roots of x2-2 a x+2=0, x2-2 b x+ 3=0 and x2-2 c x+6=0, where x1, x2, x30. Find the value of (a+b+c).

Question

Let (x1, x2),(x2, x3) and (x3, x1) are respectively the roots of x2-2 a x+2=0, x2-2 b x+ 3=0 and x2-2 c x+6=0, where x1, x2, x3>0. Find the value of (a+b+c). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/7987d7780acd13fca311e7d76a39b638-.png" /> Clearly, x 1 x 2=2, x 2 x 3=3 and x 3 x 1=6 ⇒( x 1 x 2)( x 2 x 3)( x 3 x 1)=36 or x 1 x 2 x 3=6 (As, x 1, x 2, x 3 are all positive.) So, x1=(x1 x2 x3/x2 x3)=(6/3)=2 Similarly, x2=(x1 x2 x3/x1 x3)=(6/6)=1 and x3=(x1 x2 x3/x1 x2)=(6/2)=3 Also, we have 2 a = x 1+ x 2=3 ; 2 b = x 2+ x 3=4 and 2 c = x 3+ x 1=5 Hence, 2(a+b+c)=12. ⇒( a + b + c )=6.

Q. Let (x1​,x2​),(x2​,x3​) and (x3​,x1​) are respectively the roots of x2−2ax+2=0,x2−2bx+ 3=0 and x2−2cx+6=0, where x1​,x2​,x3​>0. Find the value of (a+b+c).

Q. Let $(x_{1}, x_{2}), (x_{2}, x_{3})$ and $(x_{3}, x_{1})$ are respectively the roots of $x^{2} - 2 a x + 2 = 0, x^{2} - 2 b x +$ $3 = 0$ and $x^{2} - 2 c x + 6 = 0$ , where $x_{1}, x_{2}, x_{3} > 0$ . Find the value of $(a + b + c)$ .