The ratio of 3 number is 8:10:12.If the sum of the largest and smallest number equalls the third and 110, find the numbers
Answers
Answer:
\large\underline{\sf{Solution-}}
Solution−
Given equation is
\begin{gathered}\rm \: \frac{a}{x - b} + \frac{b}{x - a} = 2 \\ \end{gathered}
x−b
a
+
x−a
b
=2
can be further rewritten as
\begin{gathered}\rm \: \frac{a}{x - b} + \frac{b}{x - a} = 1 + 1\\ \end{gathered}
x−b
a
+
x−a
b
=1+1
can be further rearrange as
\begin{gathered}\rm \: \frac{a}{x - b} - 1 + \frac{b}{x - a} - 1 = 0\\ \end{gathered}
x−b
a
−1+
x−a
b
−1=0
\begin{gathered}\rm \: \frac{a - (x - b)}{x - b} + \frac{b - (x - a)}{x - a} = 0\\ \end{gathered}
x−b
a−(x−b)
+
x−a
b−(x−a)
=0
\begin{gathered}\rm \: \frac{a - x + b}{x - b} + \frac{b - x + a}{x - a} = 0\\ \end{gathered}
x−b
a−x+b
+
x−a
b−x+a
=0
can be rewritten as .
\begin{gathered}\rm \: \frac{a+ b - x}{x - b} + \frac{a + b - x}{x - a} = 0\\ \end{gathered}
x−b
a+b−x
+
x−a
a+b−x
=0
\begin{gathered}\rm \: (a + b - x)\bigg[\dfrac{1}{x - b} + \frac{1}{x - a} \bigg] = 0 \\ \end{gathered}
(a+b−x)[
x−b
1
+
x−a
1
]=0
\begin{gathered}\rm \: (a + b - x)\bigg[\dfrac{x - a + x - b}{(x - b)(x - a)}\bigg] = 0 \\ \end{gathered}
(a+b−x)[
(x−b)(x−a)
x−a+x−b
]=0
\begin{gathered}\rm \: (a + b - x)\bigg[\dfrac{2x - a - b}{(x - b)(x - a)}\bigg] = 0 \\ \end{gathered}
(a+b−x)[
(x−b)(x−a)
2x−a−b
]=0
\begin{gathered}\rm \: (a + b - x)(2x - a - b) = 0 \\ \end{gathered}
(a+b−x)(2x−a−b)=0
\begin{gathered}\rm\implies \: \: x = a + b \: \: \: \: \: or \: \: \: \: \: \: x = \frac{a + b}{2} \\ \end{gathered}
⟹x=a+borx=
2
a+b
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Additional Information
Nature of roots :-
Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.
If Discriminant, D > 0, then roots of the equation are real and unequal.
If Discriminant, D = 0, then roots of the equation are real and equal.
If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.
Where,
Discriminant, D = b² - 4ac