Challenge!
Topic- Coordinate Geometry
Find the minimum and maximum value of the following expression.
|\sqrt{x^{2}-26x+218}-\sqrt{x^2-2x+5}|∣
x
2
−26x+218
−
x
2
−2x+5
∣
Answers
Answer:
{\fcolorbox{aqua}{blue}{\orange{Question\:Provided\:»»}}}
QuestionProvided»»
\sqrt{(x+6)^2+5^2}+\sqrt{(x-6)^2+11^2}
(x+6)
2
+5
2
+
(x−6)
2
+11
2
{\fcolorbox{aqua}{orange}{\blue{Here's\:the\:Answer\:»»}}}
Here’stheAnswer»»
\sf{\implies\sqrt{ ( x + 6 )² + 5² } \:+\: \sqrt{ ( x - 6 )² + 11² } \:=\: 0} ⟹
(x+6)²+5²
+
(x−6)²+11²
=0
\sf{\implies\sqrt{ ( x + 6 )² + 5² } \:=\: - \sqrt{ ( x - 6 )² + 11² }}⟹
(x+6)²+5²
=−
(x−6)²+11²
\sf{\implies \bigg( \sqrt{ ( x + 6 )² + 5² } \bigg)^{2} \:=\: \bigg( - \sqrt{ ( x - 6 )² + 11² } \bigg)^{2} }⟹(
(x+6)²+5²
)
2
=(−
(x−6)²+11²
)
2
\implies \sf{ ( x + 6 )² + 5² \:=\: ( x - 6 )² + 11² } ⟹(x+6)²+5²=(x−6)²+11²
\sf{ \implies x² + 12x + 36 + 25 \:=\: x² - 12x + 36 + 121 }⟹x²+12x+36+25=x²−12x+36+121
\sf{\implies x² - x² + 12x + 12x + 61 \:=\: 36 + 121 }⟹x²−x²+12x+12x+61=36+121
\implies \sf{x² - x² + 12x + 12x + 61 \:=\: 157 } ⟹x²−x²+12x+12x+61=157
\implies \sf{ 24x + 61 \:=\: 157 }⟹24x+61=157
\implies \sf{ 24x \:=\: 157 - 61 }⟹24x=157−61
\implies \sf{ x \:=\: \dfrac{96}{24}}⟹x=
24
96
\implies \sf{\green{ x \:=\: 4 }}⟹x=4
ㅤㅤㅤ \therefore ∴ x = 4
__________________________
Formula used :
\boxed{\sf{\orange{( a + b )² \:=\: a² + 2ab + b² }}}
(a+b)²=a²+2ab+b²
\boxed{\sf{\orange{( a - b )² \:=\: a² - 2ab + b² }}}
(a−b)²=a²−2ab+b²
ㅤㅤㅤ
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