If `x=asin2t(1+cos2t)` and `y=bcos2t(1-cos2t),` show that at `pi/4,(dy)/(dx)=b/adot`

Explanation

We have given
`x=asin2t(1+cos2t)`
and `y=bcos2t(1-cos2t),`
Differentiate `x ` with respect to `t`
` frac{d x}{d t} =a[2 cos 2 t(1+cos 2 t)+sin 2 t(-2 sin 2 t)]`

`=2 a[cos 2 t+(cos ^{2} 2 t-sin ^{2} 2 t)]`
`=2 a(cos 2 t+cos 4 t)`
Differentiate `y ` with respect to `t`
`frac{d y}{d t} =b[-2 sin 2 t(1-cos 2 t)+cos 2 t(2 sin 2 t)]`
`=2 b[-sin 2 t+2 sin 2 t cos 2 t]`
`=2 b(-sin 2 t+sin 4 t)`
Therefore
`frac{d y}{d x}=frac{(frac{d y}{d t})}{(frac{d x}{d t})}=frac{2 b(-sin 2 t+sin 4 t)}{2 a(cos 2 t+cos 4 t)}`
At `t=pi/4`
`frac{d y}{d x} =frac{b(-sin frac{pi}{2}+sin pi)}{a(cos frac{pi}{2}+cos pi)}`
`=frac{b(-1+0)}{a(0-1)}`

`Rightarrow frac{d y}{d x}=frac{b}{a} `