Question
Download Solution PDFIf \(\rm\frac{\sqrt{x+20}+\sqrt{x-1}}{\sqrt{x+20}-\sqrt{x-1}}=\frac{7}{3}\), then what is the value of \(\rm \sqrt{(x + 20)(x-1)}\) ?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFGiven:
\(\rm\frac{√{x+20}+√{x-1}}{√{x+20}-√{x-1}}=\frac{7}{3}\)
Shortcut TrickAt x = 5,
\(\rm\frac{√{5+20}+√{5-1}}{√{5+20}-√{5-1}}=\frac{5+2}{5-2}=\frac{7}{3}\)
Hence, the required value
\(⇒ \rm √{(x + 20)(x-1)} = √{(25\times4)}\)
⇒ √100 = 10
Alternate Method
Concept used:
Componendo and Dividendo (reverse):
If (a + b ) : (a – b) = (c + d) : (c – d) then
a : b = c : d
Calculation:
\(\rm\frac{√{x+20}+√{x-1}}{√{x+20}-√{x-1}}=\frac{7}{3}\)
⇒ \(\rm\frac{√{x+20}}{√{x-1}}=\frac{7+3}{7-3}= \frac{10}{4}\)
⇒ \(\rm\frac{√{x+20}}{√{x-1}}=\frac{5}{2}\)
Squaring both sides, we get
⇒ \(\rm\frac{{x+20}}{{x-1}}=\frac{25}{4}\)
⇒ 4x + 80 = 25x – 25
⇒ (25x – 4x) = (80 + 25)
⇒ 21x = 105
⇒ x = 5
The value of \(\rm √{(x + 20)(x-1)}\)
⇒ \(\rm √{(5 + 20)(5-1)}\)
⇒ \(\rm √{(25\times 4)}\) = √100
⇒ \(\rm √{(x + 20)(x-1)}\) = 10
∴ The required value is 10.
Last updated on May 29, 2025
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