Doubt in common tangents

general tanegnt to y=4ax
is
y=mx+a/m
m is d slope
general tangent to
ellipse is
y=mx +(a^2 m^2 +b^2)^1/2

nw
c=a/m=(a^2m^2+b^2)^1/2

equating tehm u get m nw u get ur equation

cheers~

so ellipse is mx +√a²m²+b²,
then wat abt hyperbola?

take the derivative of both the curves
let 4 the first curve the parameter be (2t²,4t)

then u have to write the equation of the cure like

y-4t=2/t(x-2t²)

then u have to find the equation of the tangent of the 2nd curve
take a point (2 cosθ,3sinθ)
then write the equation

y-3sinθ=-3/2cotθ(x-2cosθ)

now equate the equation of both the curves to get ur point

this is the application of derivative approach

u can use the formulas also to get the answe

but i dunno rememberr