Find the equations of the tangent and the normal to the curve `x=a t^2,\ \ y=2a t` at `t=1` .

Explanation

`x=a t^{2}` and `y=2 a t`

`frac{d x}{d t}=2 a t` and `frac{d y}{d t}=2 a`

`therefore frac{d y}{d x}=frac{frac{d y}{d t}}{frac{d x}{d t}}=frac{2 a}{2 a t}=frac{1}{t}`

Slope of tangent,`m=(frac{d y}{d x})_{t=1}=frac{1}{1}=1`

Now,`(x_{1}, y_{1})=(a, 2 a)`

Equation of tangent is,


`y-y_{1}=m(x-x_{1}) `

...