if the line `xcosalpha+ysinalpha=p` is normal to the ellipse `x^2/a^2+y^2/b^2=1` then

Options

A. `a^(2)sin^(2)alpha+b^(2)cos^(2)alpha=p^(2)`
B. `a^(2)+b^(2)sin^(2)alpha=p^(2)cosec^(2)alpha`
C. `a^(2)cos^(2)alpha+b^(2)sin^(2)alpha=p^(2)`
D. None of the above

Answer

C

Explanation

Given, `x cos alpha+ysin alpha=p`
`y=-xcotalpha+p" cosec "alpha`
Since, above line is tangent to the ellipse.
`:." "c^(2)=a^(2)m^(2)+b^(2)`
`implies" "p^(2)"cosec"^(2)alpha=a^(2)cot^(2)alpha+b^(2)`
`(becausec^(2)=a^(2)m^(2)+b^(2))`
`impliesa^(2)cos^(2)alpha+b^(2)sin^(2)alpha=p^(2)`