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Question

A square with each side equal to a lies above the x-axis and has one vertex at the origin. One of the sides passing through the origin makes an angle α(0<α<π4) with the positive direction of the x-axis. Equation of a diagonal of the square is

A

y(cos αsin α)=x(sin α+cos α)

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B

y(sin α+cos α)+x(cos αsin α)=a

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C

x(cos αsin α)=y(cos α+sin α)

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D

x(cos αsin α)y(cos α+sin α)=a

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Solution

The correct options are
A

y(cos αsin α)=x(sin α+cos α)


B

y(sin α+cos α)+x(cos αsin α)=a


Let the side OA make an angle α with the x-axis. Then the coordinates of A are (a cos α, a sin α). Also, the diagonal OB makes an angle α+π4 with the x-axis, so that its equation is
y=tan(α+π4)x or y(cos αsin α)=x(sin α+cos α)

Since AC is perpendicular to OB, its slope is cot (α+π4), and as it passes through A(a cos α, a sin α), its equation is
ya sin α=cot(π4+α)(xa cos α) or y(sin α+cos α)+x(cos αsin α)=a

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