A line L has intercepts a and b on the coordinate axes. When the axes are rotated through a given angle keeping the origin fixed, this line L has the intercepts p and q . Then

Step-by-step solution

rotate the axis with respect to angle Put the values of Eq. (ii) in Eq. (i), we get aligned & x -y a + x +y b =1 & x ( a + b )+y ( b - )=1 aligned rotated axis L^ : x p + y q =1 a + b = 1 p and b - a = 1 q 1 p^2 + 1 q^2 = ( a + b )^2+ ( b - a )^2= ^2 a^2 + ^2 b^2 + 2 a b + ^2 b^2 + ^2 a^2 -2 a b = ^2 + ^2 a^2 + ^2 + ^2 b^2 1 p^2 + 1 q^2 = 1 a^2 + 1 b^2