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# ?

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Answer 1 · 2018-01-29T02:24:19

#(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#