The co-ordinate axes are rotated about the origin O in the counter-clockwise direction through an angle `60^(@)` If p and q are the intercepts made on the new axes by a straight line whose equation referred to the original axes is x + y = 1 , then `(1)/(p^(2)) + (1)/(q^(2)) = `

Options

A. 2
B. 3
C. 6
D. 8

Answer

A

Explanation

If the axes are rotated about the origin in anticlockwise direction , the relations relating old coordinates ( x, y) and new coordinates (X , Y) are
`x = X cos 60^(@) - Y sin 60^(@)` and y = `X sin 60^(@) + Y cos 60^(@)`
`implies x = (X - sqrt3 Y)/(2) + (sqrt3 X - Y)/(2) = 1 implies (sqrt3 + 1) X + (1-sqrt3) Y = 2 `
This intercepts lengths p and q on the new-coordinate axes .
`therefore p = (2)/(sqrt3 +1)` and `q = (2)/(1- sqrt3)`
`implies (1)/(p^(2)) + (1)/(q^(2)) = ((sqrt3 + 1)^(2) + (1-sqrt3)^(2))/(4) = 2`