Find the equations of tangent and normal to the curve $xy=10$ at $(2, 5)$.

The given curve is $xy=10$ and the given point $P(x, y)=(2, 5)$

diff (1) w.r.to $x$.

$x\frac{dy}{dx}+y=0\Rightarrow \frac{dy}{dx}=\frac{-y}{x}$

the slope of the tangent is $m={\left(\frac{dy}{dx}\right)}_P=\frac{-5}{2}$

the equation of the tangent at $P(2,5)$ is

$y-5=\frac{-5}{2}(x-2)$

$5x+2y-20=0$

The equation of the normal at $P(2,5)$ is

$y-5=\frac{2}{5}(x-2)\Rightarrow 5y-25=2x-4$

$\text{ i.e., }2x-5y=-21$

$\Rightarrow 2x-5y+21=0$