The correct option is
D y(cosα+sinα)+x(cosα−sinα)=aConsider the diagram shown below.
Let line OA makes an angle α with x−axis and OA=a, then coordinates of A are (acosα,asinα).
Also, OB⊥OA. Therefore, OB makes an angle (90∘+α) with x−axis, then coordinates of B are,
⇒[acos(90∘+α),asin(90∘+α)]
Here, the diagonal which is not passing through the origin is AB. Therefore, the equation of this diagonal is,
(y−asinα)=acosα−asinα−asinα−acosα(x−acosα)
(sinα+cosα)(y−asinα)=(sinα−cosα)(x−acosα)
(sinα+cosα)y−asinα(sinα+cosα)=(sinα−cosα)
y(sinα+cosα)+x(cosα−sinα)=asinα(sinα+cosα)−acosα(sinα−cosα)
y(sinα+cosα)+x(cosα−sinα)=a(sin2α+sinαcosα−cosαsinα+cos2α)
y(sinα+cosα)+x(cosα−sinα)=a
This is the required equation of the diagonal.