Show that the curves `4x=y^2` and `4x y=k` cut at right angles, if `k^2=512` .

Explanation

The given curves are,`4x=y^2.....(1)`

and `4x y=k.....(2)`
We have to prove that two curves cut at right angles if
`k^2=512`
differentiate both equation we get,
`(dy)/(dx)=2/y` and `(dy)/(dx)=-y/x`
so,`=>m_1=2/y` and `m_2=-y/x`
according to question two curves are intersecting each other with a right angle.
so, `m_1*m_2=-1`
`=>2/y*(-y/x)=-1`
`=>x=2`
now from equation `(1) and (2)`, we get ,
`y^3=k`
`=>y=k^(1/3)`
now put the value of y in eqn` (1)` we get,
`=>4x=(k^(1/3))^2`
`=>4xx2=k^(2/3)`
`=>k^2=8^3`
`=>k^2=512`