MHT CET · Maths · Hyperbola
If the line \(\mathrm{a} x+\mathrm{b} y+\mathrm{c}=0\) is a normal to the curve \(x y=1\), then
- A \(a>0, b>0\)
- B \(a>0, b < 0\)
- C \(\mathrm{a} < 0, \mathrm{~b} < 0\)
- D \(a=0, b=0\)
Answer & Solution
Correct Answer
(B) \(a>0, b < 0\)
Step-by-step Solution
Detailed explanation
\(x y=1 \)
\( \therefore y=\frac{1}{x} \)
\( \therefore y^{\prime}=\frac{-1}{x^2}\)
\(\therefore\) Slope of the normal \(=x^2\)
Slope of the line \(a x+b y+c=0\) is \(\frac{-a}{b}\).
Since the line \(a x+b y+c=0\) is a normal to the curve \(x y=1\),
\(x^2=-\frac{\mathrm{a}}{\mathrm{b}}\)
For this condition to hold true, either \(\mathrm{a} < 0, \mathrm{~b}>0\) or \(\mathrm{b} < 0, \mathrm{a}>0\)
\( \therefore y=\frac{1}{x} \)
\( \therefore y^{\prime}=\frac{-1}{x^2}\)
\(\therefore\) Slope of the normal \(=x^2\)
Slope of the line \(a x+b y+c=0\) is \(\frac{-a}{b}\).
Since the line \(a x+b y+c=0\) is a normal to the curve \(x y=1\),
\(x^2=-\frac{\mathrm{a}}{\mathrm{b}}\)
For this condition to hold true, either \(\mathrm{a} < 0, \mathrm{~b}>0\) or \(\mathrm{b} < 0, \mathrm{a}>0\)
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