Question

Find the equation of normal to the curve x = at”, y = 2at at point 't.​

Answer 1

Answer:

Equation of normal at t is

$\bf{xt+y=2at+at^3}$

Step-by-step explanation:

Find the equation of normal to the curve x = at^2, y = 2at at point 't.​

Given curve is

$x=at^2$ and $y=2at$

$\frac{dx}{dt}=2at$

$\frac{dy}{dt}=2a$

Now,

$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$

$\frac{dy}{dx}=\frac{2a}{2at}$

$\implies\:\frac{dy}{dx}=\frac{1}{t}$

Slope of tangent at 't'

$m=\frac{1}{t}$

Equation of normal at 't' is

$y-y_1=\frac{-1}{m}(x-x_1)$

$y-2at=\frac{-1}{1/t}(x-at^2)$

$y-2at=-t(x-at^2)$

$y-2at=-tx+at^3$

$xt+y=2at+at^3$

$\text{Equation of normal at t is}$

$\boxed{xt+y=2at+at^3}$