The line x cos `alpha + y sin alpha +y sin alpha =p` is tangent to the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1.`if

Options

A. `a^(2) cos ^(2) alpha-b^(2)sin^(2)alpha =p^(2)`
B. `a^(2) sin ^(2) alpha+b^(2)cos^(2)alpha =p^(2)`
C. `a^(2) cos ^(2) alpha+b^(2)sin ^(2)alpha =p^(2)`
D. `a^(2)cos^(2)alpha+b^(2) sin^(2) alpha=p`

Answer

A

Explanation

we know that the line `y=mx +c` a tangent to the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1 if c^(2) m^(2) +b^(2)`
in this case , we have
`c=-(P)/(sin alpha),m=-(cosalpha)/(sin alpha)`
so the given line will be a tangent , if
`(P^(2))/(sin^(2)alpha)=a^(2)( cos^(2)alpha)/(sin^(2)alpha)+b^(2)implies P^(2)-a^(2) cos^(2) alpha +b^(2)sin^(2) alpha `