If the distance of a given point `(alpha,beta)` from each of two straight lines `y = mx` through the origin is d, then `(alpha gamma- beta x)^(2)` is equal to (a) `x^(2) +y^(2)` (b) `d^(2)(x^(2)+y^(2))` (c) `d^(2)` (d) none of these

Options

A. `x^(2) +y^(2)`
B. `d^(2)(x^(2)+y^(2))`
C. `d^(2)`
D. none of these

Answer

B

Explanation

Any line through (0,0) is `y = mx` By given condition
`|(beta-m alpha)/(sqrt(1+m^(2)))| =d`
`(beta = m alpha)^(2) = d^(2) (1+m^(2))`
`(beta = (y//x)alpha^(2)) =d^(2) (1+(y^(2))/(x^(2)))`
`rArr ((alpha y -beta x)/(x^(2)))^(2) = d^(2) (1+y^(2)//x^(2))`
`rArr (alpha y - beta x)^(2) = d^(2) (x^(2) +y^(2))`