Question

Prove that the line `xcosA+ysinA=p` is tangent to ellipse `(x^2/a^2)+(y^2/b^2)=1` if `p^2=a^2cos^2A+b^2sin^2A` ?

Answer 1

`(acostheta,bsintheta)` represents any point on the given ellipse, `x^2/a^2+y^2/b^2=1`

The to this point on the ellipse will be

`(x*acostheta)/a^2+(y*bsintheta)/b^2=1`

`=>(xcostheta)/a+(ysintheta)/b=1`

But the given equation of the tangent is

`xcosA+ysinA=p`

`=>(xcosA)/p+(ysinA)/p=1`

Comparing these two equations of the tangent we can write

`costheta=(acosA)/p`

And

`sintheta=(bsinA)/p`

So we have

`((acosA)/p)^2+((bsinA)/p)^2=cos^2theta+sin^2theta=1`

`=>p^2=a^2cos^2A+b^2sin^2A`