Application of Derivatives 1 Question 15
15. If the line $a x+b y+c=0$ is a normal to the curve $x y=1$, then
(a) $a>0, b>0$
(b) $a>0, b<0$
$(1986,2 M)$
(c) $a<0, b>0$
(d) $a<0, b<0$
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Solution:
- Given,
$$ x y=1 \Rightarrow y=\frac{1}{x} $$
$\Rightarrow \quad \frac{d y}{d x}=-\frac{1}{x^{2}}$
Thus, slope of normal $=x^{2}$ (which is always positive) and it is given $a x+b y+c=0$ is normal, whose slope $=-\frac{a}{b}$.
$\Rightarrow \quad-\frac{a}{b}>0 \quad$ or $\quad \frac{a}{b}<0$
Hence, $a$ and $b$ are of opposite sign.
